Zimmer’s conjecture for actions of SL(m,)SL𝑚\mathrm{SL}(m,\mathbb{Z})

Aaron Brown University of Chicago, Chicago, IL 60637, USA awb@uchicago.edu David Fisher Indiana University, Bloomington, Bloomington, IN 47401, USA fisherdm@indiana.edu  and  Sebastian Hurtado University of Chicago, Chicago, IL 60637, USA shurtados@uchicago.edu
Abstract.

We prove Zimmer’s conjecture for C2superscript𝐶2C^{2} actions by finite-index subgroups of SL(m,)SL𝑚\mathrm{SL}(m,\mathbb{Z}) provided m>3𝑚3m>3. The method utilizes many ingredients from our earlier proof of the conjecture for actions by cocompact lattices in SL(m,)SL𝑚\mathrm{SL}(m,\mathbb{R}) [BFH] but new ideas are needed to overcome the lack of compactness of the space (G×M)/Γ𝐺𝑀Γ(G\times M)/\Gamma (admitting the induced G𝐺G-action). Non-compactness allows both measures and Lyapunov exponents to escape to infinity under averaging and a number of algebraic, geometric, and dynamical tools are used control this escape. New ideas are provided by the work of Lubotzky, Mozes, and Raghunathan on the structure of nonuniform lattices and, in particular, of SL(m,)SL𝑚\mathrm{SL}(m,\mathbb{Z}) providing a geometric decomposition of the cusp into rank one directions, whose geometry is more easily controlled. The proof also makes use of a precise quantitative form of non-divergence of unipotent orbits by Kleinbock and Margulis, and an extension by de la Salle of strong property (T) to representations of nonuniform lattices.

DF was partially supported by NSF Grants DMS-1308291 and DMS-1607041. DF was also partially supported by the University of Chicago, and by NSF grants DMS 1107452, 1107263, 1107367, “RNMS: Geometric Structures and Representation Varieties” (the GEAR Network) during a visit to the Isaac Newton Institute in Cambridge.

1. Introduction

1.1. Statement of results

The main result of this paper is the following:

Theorem A.

Let ΓΓ\Gamma be a finite-index subgroup of SL(m,)SL𝑚\mathrm{SL}(m,\mathbb{Z}) and let M𝑀M be a closed manifold of dimension dim(M)m2dimension𝑀𝑚2\dim(M)\leq m-2. If α:ΓDiff2(M):𝛼ΓsuperscriptDiff2𝑀\alpha\colon\Gamma\to\operatorname{Diff}^{2}(M) is a group homomorphism then α(Γ)𝛼Γ\alpha(\Gamma) is finite111After this work was completed, Brown-Damjanovic-Zhang showed that some modifications of our arguments also give a proof for C1superscript𝐶1C^{1} diffeomorphisms [BDZ].. In addition, if ω𝜔\omega is a volume form on M𝑀M, m>2𝑚2m>2 and if dim(M)m1dimension𝑀𝑚1\dim(M)\leq m-1, then if and α:ΓDiff2(M,ω):𝛼ΓsuperscriptDiff2𝑀𝜔\alpha\colon\Gamma\to\operatorname{Diff}^{2}(M,\omega) is a group homomorphism then α(Γ)𝛼Γ\alpha(\Gamma) is finite.

For m3𝑚3m\geq 3, we remark that the conclusion of Theorem A is known for actions on the circle by results of Witte Morris [Wit] (see also [Ghy, BM] for actions by more general lattices on the circle) and for volume-preserving actions on surfaces by results of Franks and Handel and of Polterovich [FH, Pol]. The proof in this paper requires that m4𝑚4m\geq 4 though we expect it can be modified to cover actions by SL(3,)SL3\mathrm{SL}(3,\mathbb{Z}); since these results are not new, we only present the case for m4𝑚4m\geq 4. While this is a very special case of Zimmer’s conjecture, it is a key example. For instance, the version of Zimmer’s conjecture restated by Margulis in his problem list [Mar2] is a special case of Theorem A.

Note that if ΓΓ\Gamma is a finite-index subgroup of SL(m,)SL𝑚\mathrm{SL}(m,\mathbb{Z}) acting on compact manifold M𝑀M, we may induce an action of SL(m,)SL𝑚\mathrm{SL}(m,\mathbb{Z}) on a (possibly non-connected) compact manifold M~=(SL(m,)×M)/\tilde{M}=(\mathrm{SL}(m,\mathbb{R})\times M)/\sim where (γ,x)(γ,x)similar-to𝛾𝑥superscript𝛾superscript𝑥(\gamma,x)\sim(\gamma^{\prime},x^{\prime}) if there is γ^Γ^𝛾Γ\hat{\gamma}\in\Gamma with γ=γγ^superscript𝛾𝛾^𝛾\gamma^{\prime}=\gamma\hat{\gamma} and x=α(γ^1)(x)superscript𝑥𝛼superscript^𝛾1𝑥x^{\prime}=\alpha(\hat{\gamma}^{-1})(x). Connectedness of M𝑀M is neither assumed nor is it used in either the proof of Theorem A or in [BFH]. Thus, for the remainder we will simply assume Γ=SL(m,)ΓSL𝑚\Gamma=\mathrm{SL}(m,\mathbb{Z}).

This paper is a first step in extending the results in [BFH] to the case where ΓΓ\Gamma is a nonuniform lattice in a split simple Lie group G𝐺G and the strategy of the proof of Theorem A relies strongly on the strategy used in [BFH]. In the remainder of the introduction, we recall the proof in the cocompact case, indicate where the difficulties arise in the nonuniform case, and outline the proof of Theorem A. At the end of the introduction we make some remarks on other approaches and difficulties we encountered.

We recall a key definition from [BFH]. Let ΓΓ\Gamma be a finitely generated group. Let :Γ:Γ\ell\colon\Gamma\to\mathbb{N} denote the word-length function with respect to some choice of finite generating set for ΓΓ\Gamma. Given a C1superscript𝐶1C^{1} diffeomorphism f:MM:𝑓𝑀𝑀f\colon M\to M let Df=supxMDxfnorm𝐷𝑓subscriptsupremum𝑥𝑀normsubscript𝐷𝑥𝑓\|Df\|=\sup_{x\in M}\|D_{x}f\| (for some choice of norm on TM𝑇𝑀TM).

Definition 1.1.

An action α:ΓDiff1(M):𝛼ΓsuperscriptDiff1𝑀\alpha\colon\Gamma\to\mathrm{Diff}^{1}(M) has uniform subexponential growth of derivatives if

for every ε>0𝜀0\varepsilon>0, there is Cεsubscript𝐶𝜀C_{\varepsilon} such that Dα(γ)Cεeε(γ)norm𝐷𝛼𝛾subscript𝐶𝜀superscript𝑒𝜀𝛾\|D\alpha(\gamma)\|\leq C_{\varepsilon}e^{\varepsilon\ell(\gamma)} for all γΓ.𝛾Γ\gamma\in\Gamma. (1)

The main result of the paper is the following:

Theorem B.

For m4𝑚4m\geq 4, let Γ=SL(m,)ΓSL𝑚\Gamma=\mathrm{SL}(m,\mathbb{Z}) and let M𝑀M be a closed manifold.

  1. (1)

    If dim(M)m2dimension𝑀𝑚2\dim(M)\leq m-2 then any action α:ΓDiff2(M):𝛼ΓsuperscriptDiff2𝑀\alpha\colon\Gamma\to\operatorname{Diff}^{2}(M) has uniform subexponential growth of derivatives;

  2. (2)

    if ω𝜔\omega is a volume form on M𝑀M and dim(M)m1dimension𝑀𝑚1\dim(M)\leq m-1 then any action α:ΓDiff2(M,ω):𝛼ΓsuperscriptDiff2𝑀𝜔\alpha\colon\Gamma\to\operatorname{Diff}^{2}(M,\omega) has uniform subexponential growth of derivatives.

To deduce Theorem A from Theorem B, we apply [BFH, Theorem 2.9] and de la Salle’s recent result establishing strong property (T)𝑇(T) for nonuniform lattices [dlS, Theorem 1.2] and conclude that any action α𝛼\alpha as in Theorem A preserves a continuous Riemannian metric. For clarity, we point out that we need de la Salle’s Theorem 1.21.21.2 and not his Theorem 1.11.11.1 because we need the measures converging to the projection to be positive measures. That Theorem [dlS, Theorem 1.2] provides positive measures where [dlS, Theorem 1.1] does not is further clarified in [dlS, Section 2.3]. Once a continuous invariant metric is preserved, the image of any homomorphism α𝛼\alpha in Theorem A is contained in a compact Lie group K𝐾K. All such homomorphisms necessarily have finite image due to the presence of unipotent elements in SL(m,)SL𝑚\mathrm{SL}(m,\mathbb{Z}). We remark that while the finiteness of the image of α𝛼\alpha was deduced using Margulis’s superrigidity theorem in [BFH], it is unnecessary in the setting of Theorem A since, as any unipotent element of SL(m,)SL𝑚\mathrm{SL}(m,\mathbb{Z}) lies in the center of some integral Heisenberg subgroup of SL(m,)SL𝑚\mathrm{SL}(m,\mathbb{Z}), all unipotent elements have finite image in K𝐾K and therefore so does SL(m,)SL𝑚\mathrm{SL}(m,\mathbb{Z}).

1.2. Review of the cocompact case

To explain the proof of Theorem B, we briefly explain the difficulties in extending the arguments from [BFH] to the setting of actions by nonuniform lattices. We begin by recalling the proof in the cocompact setting.

In both [BFH] and the proof of Theorem B, we consider a fiber bundle

MMα:=(G×M)/Γ𝜋G/Γ𝑀superscript𝑀𝛼assign𝐺𝑀Γ𝜋𝐺ΓM\rightarrow M^{\alpha}:=(G\times M)/\Gamma\xrightarrow{\pi}G/\Gamma

which allows us to replace the ΓΓ\Gamma-action on M𝑀M with a G𝐺G-action on Mαsuperscript𝑀𝛼M^{\alpha}. In the case that ΓΓ\Gamma is cocompact, showing subexponential growth of derivatives of the ΓΓ\Gamma-action is equivalent to showing subexponential growth of the fiberwise derivative cocycle for the G𝐺G-action.

To prove such subexponential growth for the G𝐺G-action on Mαsuperscript𝑀𝛼M^{\alpha} we argued by contradiction to obtain a sequence of points xnMαsubscript𝑥𝑛superscript𝑀𝛼x_{n}\in M^{\alpha} and semisimple elements ansubscript𝑎𝑛a_{n} in a Cartan subgroup AG𝐴𝐺A\subset G which satisfy Dxnan|Feλd(an,Id)\|{D_{x_{n}}a_{n}}{|_{{F}}}\|\geq e^{\lambda d(a_{n},\operatorname{Id})} for some λ>0𝜆0\lambda>0. Here Dxgsubscript𝐷𝑥𝑔D_{x}g denotes the derivative of translation by g𝑔g at xMα𝑥superscript𝑀𝛼x\in M^{\alpha}, F𝐹F is the fiberwise tangent bundle of Mαsuperscript𝑀𝛼M^{\alpha}, and Dxnan|Fevaluated-atsubscript𝐷subscript𝑥𝑛subscript𝑎𝑛𝐹{D_{x_{n}}a_{n}}{|_{{F}}} is the restriction of Dxnansubscript𝐷subscript𝑥𝑛subscript𝑎𝑛D_{x_{n}}a_{n} to F(xn)𝐹subscript𝑥𝑛{F(x_{n})}.

The pairs (xn,an)subscript𝑥𝑛subscript𝑎𝑛(x_{n},a_{n}) determine empirical measures μnsubscript𝜇𝑛\mu_{n} on Mαsuperscript𝑀𝛼M^{\alpha} supported on the orbit {ans(xn):0stn}conditional-setsuperscriptsubscript𝑎𝑛𝑠subscript𝑥𝑛0𝑠subscript𝑡𝑛\{a_{n}^{s}(x_{n}):0\leq s\leq t_{n}\} which accumulate on a measure μ𝜇\mu that is a𝑎a-invariant for some aA𝑎𝐴a\in A and has a positive Lyapunov exponent for the fiberwise derivative cocycle of size at least λ𝜆\lambda. Using classical results in homogeneous dynamics in conjunction with the key proposition from [BRHW], we averaged the measure μ𝜇\mu to obtain a G𝐺G-invariant measure μsuperscript𝜇\mu^{\prime} on Mαsuperscript𝑀𝛼M^{\alpha} with a non-zero fiberwise Lyapunov exponent; the existence of such a measure μsuperscript𝜇\mu^{\prime} contradicts Zimmer’s cocycle superrigidity theorem.

1.3. Difficulties in the nonuniform setting.

When ΓΓ\Gamma is nonuniform the space Mαsuperscript𝑀𝛼M^{\alpha} is not compact and the sequence of empirical measures μnsubscript𝜇𝑛\mu_{n} might diverge to infinity in Mαsuperscript𝑀𝛼M^{\alpha}; that is, in the limit we might have a “loss of mass”. Additionally, even if the measures {μn}subscript𝜇𝑛\{\mu_{n}\} satisfy some tightness criteria so as to prevent escape of mass, one might have “escape of Lyapunov exponents:” for a limiting measure μ𝜇\mu, the Lyapunov exponents may be infinite or the value could drop below the value expected by the growth of fiberwise cocycles along the orbits {as(xn):0stn}conditional-setsuperscript𝑎𝑠subscript𝑥𝑛0𝑠subscript𝑡𝑛\{a^{s}(x_{n}):0\leq s\leq t_{n}\}. For instance, the contribution to the exponential growth of derivatives along the sequence of empirical measures could arise primarily from excursions of orbits deep into the cusp. If one makes naïve computations with the return cocycle β:G×G/ΓΓ:𝛽𝐺𝐺ΓΓ\beta\colon G\times G/\Gamma\rightarrow\Gamma (measuring for x𝑥x in a fundamental domain D𝐷D the element of ΓΓ\Gamma needed to bring gx𝑔𝑥gx back to a D𝐷D) one in fact expects that the fiberwise derivative are very large for translations of points far out in the cusp since the orbits of such points cross a large number of fundamental domains. The weakest consequence of this observation is that subexponential growth of the fiberwise derivative of the induced G𝐺G-action is much stronger than subexponential growth of derivatives of the ΓΓ\Gamma-action. While we still work with the induced G𝐺G-action and the fiberwise derivative in many places, the arguments become more complicated than in the cocompact case.

In the homogeneous dynamics literature, there are many tools to study escape of mass. Controlling the escape of Lyapunov exponents seems to be more novel. To rule out escape of mass, it suffices to prove tightness of family of measures {μn}.subscript𝜇𝑛\{\mu_{n}\}. To control Lyapunov exponents, we introduce a quantitative tightness condition: we construct measures {μn}subscript𝜇𝑛\{\mu_{n}\} with uniformly exponentially small mass in the cusps. See Section 3.2. It is a standard computation to show the Haar measure on SL(m,)/SL(m,)SL𝑚SL𝑚\mathrm{SL}(m,\mathbb{R})/\mathrm{SL}(m,\mathbb{Z}) (or any G/Γ𝐺ΓG/\Gamma where G𝐺G is semisimple and ΓΓ\Gamma is a lattice) has exponentially small mass in the cusps.

1.4. Outline of proof

With the above difficulties in mind, we outline the strategy of the proof of Theorem B. Lubotzky, Mozes and Raghunathan proved that SL(m,)SL𝑚\mathrm{SL}(m,\mathbb{Z}) is quasi-isometrically embedded in SL(m,)SL𝑚\mathrm{SL}(m,\mathbb{R}). And in this special case, they give a proof that every element γSL(m,)𝛾SL𝑚\gamma\in\mathrm{SL}(m,\mathbb{Z}) can be written as a product of at most m2superscript𝑚2m^{2} elements δisubscript𝛿𝑖\delta_{i} contained in canonical copies of SL(2,)SL2\mathrm{SL}(2,\mathbb{Z}) determined by pairs of standard basis vectors for msuperscript𝑚\mathbb{R}^{m}; moreover the word-length of each δisubscript𝛿𝑖\delta_{i} is at most proportional to the word-length of γ𝛾\gamma [LMR1, Corollary 3]. (We note however that such effective generation of ΓΓ\Gamma only holds for SL(m,)SL𝑚\mathrm{SL}(m,\mathbb{Z}); for the general case, in [LMR2] a weaker generation of ΓΓ\Gamma in terms of \mathbb{Q}-rank 1 subgroups is shown.) Thus, to show uniform subexponential growth of derivatives for the action of SL(m,)SL𝑚\mathrm{SL}(m,\mathbb{Z}), it suffices to show uniform subexponential growth of derivatives for the restriction of our action to each canonical copy of SL(2,)SL2\mathrm{SL}(2,\mathbb{Z}).

We first obtain uniform subexponential growth of derivatives for the unipotent elements in SL(2,)SL2\mathrm{SL}(2,\mathbb{Z}) in Section 4. See Proposition 4.1. The strategy is to consider a subgroup of the form SL(2,)2SL(m,)left-normal-factor-semidirect-productSL2superscript2SL𝑚\mathrm{SL}(2,\mathbb{Z})\ltimes\mathbb{Z}^{2}\subset\mathrm{SL}(m,\mathbb{Z}). We first prove that a large proportion of elements in SL(2,)SL2\mathrm{SL}(2,\mathbb{Z}) satisfy (1). To prove this, we use that if at:=diag(et,et)assignsuperscript𝑎𝑡diagsuperscript𝑒𝑡superscript𝑒𝑡a^{t}:=\text{diag}(e^{t},e^{-t}) then a typical atsuperscript𝑎𝑡a^{t}-orbit in SL(2,)/SL(2,)SL2SL2\mathrm{SL}(2,\mathbb{R})/\mathrm{SL}(2,\mathbb{Z}) equidistributes to the Haar measure. In particular, for the empirical measures along such a𝑎a-orbits we apply the techniques from [BFH] to show subexponential growth of fiberwise derivatives along such orbits and conclude that a large proportion of SL(2,)SL2\mathrm{SL}(2,\mathbb{Z}) satisfies (1). See Proposition 4.2. The proof of this fact repeats most of the ideas and techniques from [BFH] as well a quantitative non-divergence of unipotent averages following Kleinbock and Margulis. The precise averaging procedure is different here than in [BFH].

Having shown Proposition 4.2, we consider the SL(2,)SL2\mathrm{SL}(2,\mathbb{Z})-action on the normal subgroup 2superscript2\mathbb{Z}^{2} of SL(2,)2left-normal-factor-semidirect-productSL2superscript2\mathrm{SL}(2,\mathbb{Z})\ltimes\mathbb{Z}^{2} to show that for every n0𝑛0n\geq 0, the ball Bnsubscript𝐵𝑛B_{n} of radius n𝑛n in 2superscript2\mathbb{Z}^{2} contains a positive-density subset of unipotent elements satisfying (1). Taking iterated sumsets of such good unipotent elements of Bn(2)subscript𝐵𝑛superscript2B_{n}(\mathbb{Z}^{2}) with a finite set one obtains uniform subexponential growth of derivatives for every element in Bnsubscript𝐵𝑛B_{n}. This relies heavily on the fact that 2superscript2\mathbb{Z}^{2} is abelian. See Subsection 4.2.

It is worth noting that the subgroups of the form SL(2,)2Γleft-normal-factor-semidirect-productSL2superscript2Γ\mathrm{SL}(2,\mathbb{Z})\ltimes\mathbb{Z}^{2}\subset\Gamma are also considered in the work of Lubotzky, Mozes, and Raghunathan in [LMR1] as well as in Margulis’s early constructions of expander graphs and subsequent work on property (T) and expanders [Mar1].

Having established Proposition 4.1, we assume for the sake of contradiction that the restriction of α𝛼\alpha to SL(2,)SL2\mathrm{SL}(2,\mathbb{Z}) fails to exhibit uniform subexponential growth of derivatives. We obtain in Subsection 5.2 a sequence ζnsubscript𝜁𝑛\zeta_{n} of atsuperscript𝑎𝑡a^{t}-orbit segments in SL(2,)/SL(2,)SL2SL2\mathrm{SL}(2,\mathbb{R})/\mathrm{SL}(2,\mathbb{Z}) which drift only a sub-linear distance into the cusp with respect to their length and accumulate exponential growth of the fiberwise derivative. Here we use that orbits deep in the cusp of SL(2,)/SL(2,)SL2SL2\mathrm{SL}(2,\mathbb{R})/\mathrm{SL}(2,\mathbb{Z}) correspond to unipotent deck transformations and that Proposition 4.1 implies that these do not contribute to the exponential growth of the fiberwise derivative. Here, we heavily use the structure of SL(2,)SL2\mathrm{SL}(2,\mathbb{Z}) subgroups.

We promote the family of orbit segments ζnsubscript𝜁𝑛\zeta_{n} in Mαsuperscript𝑀𝛼M^{\alpha} to a family of measures {μn}subscript𝜇𝑛\{\mu_{n}\} all of whose subsequential limits are A𝐴A-invariant measures μ𝜇\mu on Mαsuperscript𝑀𝛼M^{\alpha} with non-zero fiberwise exponents. To construct μnsubscript𝜇𝑛\mu_{n}, we construct a Følner sequence FnGsubscript𝐹𝑛𝐺F_{n}\subset G inside a solvable subgroup AN𝐴superscript𝑁AN^{\prime} where A𝐴A is the full Cartan subgroup of SL(m,)SL𝑚\mathrm{SL}(m,\mathbb{R}) and Nsuperscript𝑁N^{\prime} is a well-chosen abelian subgroup of unipotent elements. We average our orbit segments ζnsubscript𝜁𝑛\zeta_{n} over Fnsubscript𝐹𝑛F_{n} to obtain the sequence of measures μnsubscript𝜇𝑛\mu_{n} in Mαsuperscript𝑀𝛼M^{\alpha}. In general, Følner sets for AN𝐴superscript𝑁AN^{\prime} are subsets which are linearly large in the A𝐴A-direction and exponentially large in the Nsuperscript𝑁N^{\prime} direction. In our case the Nsuperscript𝑁N^{\prime}-part will not affect the Lyapunov exponent because we work inside a subset where the return cocycle β𝛽\beta restricted to Nsuperscript𝑁N^{\prime} takes unipotent values and we have already proven subexponential growth of the fiberwise derivatives for unipotent elements.

The fact that μnsubscript𝜇𝑛\mu_{n} behaves well in the cusp is due to two facts: First, the segments obtained in Subsection 5.2 do not drift too deep into the cusp of SL(2,)/SL(2,)SL2SL2\mathrm{SL}(2,\mathbb{R})/\mathrm{SL}(2,\mathbb{Z}). Second, we choose our subgroup Nsuperscript𝑁N^{\prime} such that the Nsuperscript𝑁N^{\prime}-orbits of each point along each ζnsubscript𝜁𝑛\zeta_{n} is a closed torus that is well-behaved when translated by A𝐴A. The argument here is related to the fact closed horocycles in the cusp of SL(2,)/SL(2,)SL2SL2\mathrm{SL}(2,\mathbb{R})/\mathrm{SL}(2,\mathbb{Z}) equidistribute to the Haar measure when flowed backwards by the geodesic flow.

To finish the argument, we show that any AN𝐴superscript𝑁AN^{\prime}-invariant measure on Mαsuperscript𝑀𝛼M^{\alpha} projects to Haar measure on SL(m,)/SL(m,)SL𝑚SL𝑚\mathrm{SL}(m,\mathbb{R})/\mathrm{SL}(m,\mathbb{Z}) using Ratner’s measure classification and equidistribution theorems. Then, as in [BFH], we can use [BRHW, Proposition 5.1] and argue as in the cocompact case in [BFH] show that μ𝜇\mu is in fact G𝐺G-invariant and thereby obtain a contradiction with Zimmer’s cocycle superrigidity theorem.

1.5. A few remarks on other approaches.

We close the introduction by making some remarks on other approaches, particularly other approaches for controlling the escape of mass. We emphasize here that one key difficulty for all approaches is that we are not able to control the “images” of the cocycle β:G×G/ΓΓ:𝛽𝐺𝐺ΓΓ\beta\colon G\times G/\Gamma\rightarrow\Gamma in either our special case or in general. To understand this remark better, consider first the case where G=SL(2,)𝐺SL2G=\mathrm{SL}(2,\mathbb{R}) and Γ=SL(2,)ΓSL2\Gamma=\mathrm{SL}(2,\mathbb{Z}). If we take a one-parameter subgroup c(t)<SL(2,)𝑐𝑡SL2c(t)<\mathrm{SL}(2,\mathbb{R}) and take the trajectory c(t)x𝑐𝑡𝑥c(t)x for t𝑡t in some interval [0,T]0𝑇[0,T] and assume and assume the entire trajectory on G/Γ𝐺ΓG/\Gamma lies deep enough in the cusp, then β(a(t),x)𝛽𝑎𝑡𝑥\beta(a(t),x) is necessarily unipotent for all t𝑡t in [0,T]0𝑇[0,T]. No similar statement is true for G=SL(m,)𝐺SL𝑚G=\mathrm{SL}(m,\mathbb{R}) and Γ=SL(m,)ΓSL𝑚\Gamma=\mathrm{SL}(m,\mathbb{Z}). In fact analogous statements are true if and only if ΓΓ\Gamma has \mathbb{Q}-rank one, this is closely related to the fact that higher \mathbb{Q}-rank locally symmetric spaces are 111-connected at infinity. This forces us to “factor” the action into actions of rank-one subgroups in order to control the growth of derivatives.

One might hope to obtain subexponential growth of derivatives more directly for all elements of SL(2,)SL2\mathrm{SL}(2,\mathbb{Z}), or even directly in SL(m,)SL𝑚\mathrm{SL}(m,\mathbb{Z}), by proving better estimates on the size of the “generic” subsets of SL(2,)SL2\mathrm{SL}(2,\mathbb{R}) (or SL(m,)SL𝑚\mathrm{SL}(m,\mathbb{R})) whose A𝐴A-orbits define empirical measures satisfying some tightness condition. While one can get good estimates on the size of the sets in Proposition 4.2 using Margulis functions and large deviation estimates as in [Ath, EM2], the resulting estimates are not sharp enough to allow us to prove subexponential growth of derivatives. One can compare with the conjectures in [KKLM] about loss of mass.

An elementary related question is the following: Let Bnsubscript𝐵𝑛B_{n} be a ball of radius n𝑛n in a Lie group G𝐺G (or a lattice ΓΓ\Gamma) and suppose there exists subset Snsubscript𝑆𝑛S_{n} of Bnsubscript𝐵𝑛B_{n} such that Snsubscript𝑆𝑛S_{n} and Bnsubscript𝐵𝑛B_{n} have more or less equal mass, meaning that:

vol(BnSn)vol(Bn)<εn𝑣𝑜𝑙subscript𝐵𝑛subscript𝑆𝑛𝑣𝑜𝑙subscript𝐵𝑛subscript𝜀𝑛\frac{vol(B_{n}\setminus S_{n})}{vol(B_{n})}<\varepsilon_{n}

for a certain sequence εnsubscript𝜀𝑛\varepsilon_{n} of numbers converging to zero. Does there exists an integer k𝑘k (independent of n𝑛n) such that for n𝑛n large:

BnSnSnkSnB_{n}\subset S_{n}*S_{n}*\stackrel{{\scriptstyle k}}{{\cdots}}*S_{n} (2)

Observe that the question depends on how fast εnsubscript𝜀𝑛\varepsilon_{n} is decreasing and on the group G𝐺G. For example if G𝐺G abelian, εnsubscript𝜀𝑛\varepsilon_{n} can be a sufficiently small constant as a consequence of Proposition 4.9. Also, it is not hard to see that for any group G𝐺G the existence of k𝑘k is guaranteed if εnsubscript𝜀𝑛\varepsilon_{n} decreases exponentially quickly. So the real question is how fast εnsubscript𝜀𝑛\varepsilon_{n} has to decrease to zero in order for this statement to hold. Does (2) holds for G=SL3()𝐺subscriptSL3G=\mathrm{SL}_{3}(\mathbb{Z}) and εn=2ncsubscript𝜀𝑛superscript2superscript𝑛𝑐\varepsilon_{n}=2^{-n^{c}} for some c<1𝑐1c<1? If the answer to this question is yes, then it would be possible to approach our results via Margulis functions and large deviation estimates.

Acknowledgements

We thank Dave Witte Morris for his generous willingness to answer questions of all sorts throughout the production of this paper and [BFH]. We also thank to Shirali Kadyrov, Jayadev Athreya and Alex Eskin for helpful conversations, particularly on the material in Subsection 1.5 and Mikael de la Salle for many helpful conversations regarding strong property (T)𝑇(T). We also thank the anonymous referee for a very careful reading and numerous comments which helped improve the exposition.

2. Standing notation

We review the notation introduced in [BFH] and establish some standing notation and conventions as well as state some facts used in the remainder of the paper.

2.1. Lie theoretic and geometric notation

We write G=SL(m,)𝐺SL𝑚G=\mathrm{SL}(m,\mathbb{R}) and Γ=SL(m,)ΓSL𝑚\Gamma=\mathrm{SL}(m,\mathbb{Z}). Let 𝔤𝔤\mathfrak{g} denote the Lie algebra of G𝐺G. Let IdId\operatorname{Id} denote the identity element of G𝐺G. We fix the standard Cartan involution θ:𝔤𝔤:𝜃𝔤𝔤\theta\colon\mathfrak{g}\to\mathfrak{g} given by θ(X)=Xt𝜃𝑋superscript𝑋𝑡\theta(X)=-X^{t} and write 𝔨𝔨\mathfrak{k} and 𝔭𝔭\mathfrak{p}, respectively, for the +11+1 and 11-1 eigenspaces of θ𝜃\theta. Define 𝔞𝔞\mathfrak{a} to be a maximal abelian subalgebra of 𝔭𝔭\mathfrak{p}. Then 𝔞𝔞\mathfrak{a} is the vector space of diagonal matrices.

The roots of 𝔤𝔤\mathfrak{g} are the linear functionals βi,j𝔞subscript𝛽𝑖𝑗superscript𝔞\beta_{i,j}\in\mathfrak{a}^{*} defined as

βi,j(diag(t1,,tm))=titj.subscript𝛽𝑖𝑗diagsubscript𝑡1subscript𝑡𝑚subscript𝑡𝑖subscript𝑡𝑗\beta_{i,j}(\mathrm{diag}(t_{1},\dots,t_{m}))=t_{i}-t_{j}.

The simple positive roots are αj=βj,j+1subscript𝛼𝑗subscript𝛽𝑗𝑗1\alpha_{j}=\beta_{j,j+1} and the positive roots are the positive integral combinations of {αj}subscript𝛼𝑗\{\alpha_{j}\} that are still roots.

For a root β𝛽\beta, write 𝔤βsuperscript𝔤𝛽\mathfrak{g}^{\beta} for the associated root space. Each root space 𝔤βsuperscript𝔤𝛽\mathfrak{g}^{\beta} exponentiates to a 1-parameter unipotent subgroup UβGsuperscript𝑈𝛽𝐺U^{\beta}\subset G. The Lie subalgebra 𝔫𝔫\mathfrak{n} generated by all root spaces 𝔤βsuperscript𝔤𝛽\mathfrak{g}^{\beta} for positive roots β𝛽\beta, coincides with the Lie algebra of all strictly upper-triangular matrices.

Let A,N,𝐴𝑁A,N, and K𝐾K be the analytic subgroups of G𝐺G corresponding to 𝔞,𝔫𝔞𝔫\mathfrak{a},\mathfrak{n} and 𝔨𝔨\mathfrak{k}. Then

  1. (1)

    A=exp(𝔞)𝐴𝔞A=\exp(\mathfrak{a}) is the group of all diagonal matrices with positive entries. A𝐴A is an abelian group and we identity linear functionals on 𝔞𝔞\mathfrak{a} with linear functionals on A𝐴A via the exponential map exp:𝔞A:𝔞𝐴\exp\colon\mathfrak{a}\to A;

  2. (2)

    N=exp(𝔫)𝑁𝔫N=\exp(\mathfrak{n}) is the group of upper-triangular matrices with 111s on the diagonal;

  3. (3)

    K=SO(m)𝐾SO𝑚K=\mathrm{SO}(m).

The Weyl group of G𝐺G is the group of permutation matrices. This acts transitively on the set of all roots ΣΣ\Sigma.

For 1i,jmformulae-sequence1𝑖𝑗𝑚1\leq i,j\leq m, the subgroup of G𝐺G generated by Uβi,jsuperscript𝑈subscript𝛽𝑖𝑗U^{\beta_{i,j}} and Uβj,isuperscript𝑈subscript𝛽𝑗𝑖U^{\beta_{j,i}} is isomorphic to SL(2,)SL2\mathrm{SL}(2,\mathbb{R}). We denote this subgroup by Hi,j=SLi,j(2,).subscript𝐻𝑖𝑗subscriptSL𝑖𝑗2H_{i,j}=\mathrm{SL}_{{i,j}}(2,\mathbb{R}). Then Λi,j:=Hi,jΓassignsubscriptΛ𝑖𝑗subscript𝐻𝑖𝑗Γ\Lambda_{i,j}:=H_{i,j}\cap\Gamma is a lattice in SLi,j(2,)subscriptSL𝑖𝑗2\mathrm{SL}_{{i,j}}(2,\mathbb{R}) isomorphic to SL(2,)SL2\mathrm{SL}(2,\mathbb{Z}). Note then that Xi,j:=Hi,j/Λi,jassignsubscript𝑋𝑖𝑗subscript𝐻𝑖𝑗subscriptΛ𝑖𝑗X_{i,j}:=H_{i,j}/\Lambda_{i,j} is the unit tangent bundle to the modular surface. We will use the standard notation Ei,jsubscript𝐸𝑖𝑗E_{i,j} for an elementary matrix with 1s on the diagonal and in the (i,j)𝑖𝑗(i,j)-place and 0s everywhere else. Note that Ei,jsubscript𝐸𝑖𝑗E_{i,j} and Ej,isubscript𝐸𝑗𝑖E_{j,i} generate Λi,jsubscriptΛ𝑖𝑗\Lambda_{i,j}.

We equip G𝐺G with a left-K𝐾K-invariant and right-G𝐺G-invariant metric. Such a metric is unique up to scaling. Let d𝑑d denote be the induced distance on G𝐺G. With respect to this metric and distance d𝑑d, each Hi,jsubscript𝐻𝑖𝑗H_{i,j} is geodesically embedded. By rescaling the metric, we may assume the restriction of d𝑑d to each Hi,jsubscript𝐻𝑖𝑗H_{i,j} coincides with the standard metric of constant curvature 11-1 on the upper half plane SO(2)\SL(2,)\SO2SL2\mathrm{SO}(2)\backslash\mathrm{SL}(2,\mathbb{R}). This metric has the following properties that we exploit throughout.

  1. (1)

    For any matrix norm \|\cdot\| on Hi,jSL(2,)similar-to-or-equalssubscript𝐻𝑖𝑗SL2H_{i,j}\simeq\mathrm{SL}(2,\mathbb{R}) there is a C1subscript𝐶1C_{1} such that

    2logAC1d(A,Id)2logA+C12norm𝐴subscript𝐶1𝑑𝐴Id2norm𝐴subscript𝐶1{2}\log\|A\|-C_{1}\leq d(A,\operatorname{Id})\leq{2}\log\|A\|+C_{1} (3)

    for all AHi,j𝐴subscript𝐻𝑖𝑗A\in H_{i,j}.

  2. (2)

    Let B(Id,r)𝐵Id𝑟B(\operatorname{Id},r) denote the metric ball of radius r𝑟r in Hi,jsubscript𝐻𝑖𝑗H_{i,j} centered at IdId\mathrm{Id}. Then with respect to the induced Riemannian volume on Hi,jsubscript𝐻𝑖𝑗H_{i,j} we have

    vol(B(Id,r))=4π(cosh(r)1)4πervol𝐵Id𝑟4𝜋𝑟14𝜋superscript𝑒𝑟\mathrm{vol}(B(\mathrm{Id},r))=4\pi(\cosh(r)-1)\leq 4\pi e^{r}

    and for all sufficiently large r>0𝑟0r>0

    vol(B(x,r))er.vol𝐵𝑥𝑟superscript𝑒𝑟\mathrm{vol}(B(x,r))\geq e^{r}. (4)
  3. (3)

    For any matrix norm \|\cdot\| on SL(m,)SL𝑚\mathrm{SL}(m,\mathbb{R}), there are constants C0>1subscript𝐶01C_{0}>1 and κ>1𝜅1\kappa>1 such that for any matrix ASL(m,)𝐴SL𝑚A\in\mathrm{SL}(m,\mathbb{R}) we have

    κ1logAC0d(A,Id)κlogA+C0.superscript𝜅1delimited-∥∥𝐴subscript𝐶0𝑑𝐴Id𝜅delimited-∥∥𝐴subscript𝐶0\begin{gathered}\kappa^{-1}\log\|A\|-C_{0}\leq d(A,\mathrm{Id})\leq\kappa\log\|A\|+C_{0}.\end{gathered} (5)
  4. (4)

    In particular, there are C2subscript𝐶2C_{2} and C3subscript𝐶3C_{3} so that if Ei,jSL(m,)subscript𝐸𝑖𝑗SL𝑚E_{i,j}\in\mathrm{SL}(m,\mathbb{Z}) is an elementary unipotent matrix then

    d(Ei,jk,Id)C2logk+C3.𝑑superscriptsubscript𝐸𝑖𝑗𝑘Idsubscript𝐶2𝑘subscript𝐶3d(E_{i,j}^{k},\mathrm{Id})\leq C_{2}\log k+C_{3}. (6)

2.2. Suspension space and induced G𝐺G-action

Let Mα=(G×M)/Γsuperscript𝑀𝛼𝐺𝑀ΓM^{\alpha}=(G\times M)/\Gamma be the fiber-bundle over SL(m,)/SL(m,)SL𝑚SL𝑚\mathrm{SL}(m,\mathbb{R})/\mathrm{SL}(m,\mathbb{Z}) obtained as follows: on G×M𝐺𝑀G\times M let ΓΓ\Gamma act as

(g,x)γ=(gγ,α(γ1)(x))𝑔𝑥𝛾𝑔𝛾𝛼superscript𝛾1𝑥(g,x)\cdot\gamma=(g\gamma,\alpha(\gamma^{-1})(x))

and let G𝐺G act as

g(g,x)=(gg,x).superscript𝑔𝑔𝑥superscript𝑔𝑔𝑥g^{\prime}\cdot(g,x)=(g^{\prime}g,x).

The G𝐺G-action on G×M𝐺𝑀G\times M descends to a G𝐺G-action on the quotient Mα=(G×M)/Γsuperscript𝑀𝛼𝐺𝑀ΓM^{\alpha}=(G\times M)/\Gamma. Let π:MαSL(m,)/SL(m,):𝜋superscript𝑀𝛼SL𝑚SL𝑚\pi\colon M^{\alpha}\to\mathrm{SL}(m,\mathbb{R})/\mathrm{SL}(m,\mathbb{Z}) be the canonical projection. As in [BFH], we write F=kerDπ𝐹kernel𝐷𝜋F=\ker D\pi for the fiberwise tangent bundle to Mαsuperscript𝑀𝛼M^{\alpha}. Write F𝐹\mathbb{P}F for the projectivization of the fiberwise tangent bundle. We write Dxg|F:F(x)F(gx):evaluated-atsubscript𝐷𝑥𝑔𝐹𝐹𝑥𝐹𝑔𝑥{D_{x}g}{|_{{F}}}\colon F(x)\rightarrow F(gx) for the fiberwise derivative as in [BFH]. For (x,[v])F𝑥delimited-[]𝑣𝐹(x,[v])\in\mathbb{P}F and gG𝑔𝐺g\in G, write

g(x,[v]):=(gx,[Dxg|F(x)v])assign𝑔𝑥delimited-[]𝑣𝑔𝑥delimited-[]evaluated-atsubscript𝐷𝑥𝑔𝐹𝑥𝑣g\cdot(x,[v]):=(g\cdot x,[{D_{x}g}{|_{{F(x)}}}v])

for the action of g𝑔g on F𝐹\mathbb{P}F induced by Dxg|Fevaluated-atsubscript𝐷𝑥𝑔𝐹{D_{x}g}{|_{{F}}}.

We follow [BRHW, Section 2.1] and equip G×M𝐺𝑀G\times M with a C1superscript𝐶1C^{1} Riemannian metric ,\langle\cdot,\cdot\rangle with the following properties:

  1. (1)

    ,\langle\cdot,\cdot\rangle is ΓΓ\Gamma-invariant.

  2. (2)

    for xM𝑥𝑀x\in M and gG𝑔𝐺g\in G, under the canonical identification of the G𝐺G-orbit of (g,x)𝑔𝑥(g,x) with G𝐺G, the restriction of ,\langle\cdot,\cdot\rangle to the G𝐺G-orbit of (g,x)𝑔𝑥(g,x) coincides with the fixed right-invariant metric on G𝐺G.

  3. (3)

    There is a Siegel fundamental set DG𝐷𝐺D\subset G and C>1𝐶1C>1 such that for any g1,g2Dsubscript𝑔1subscript𝑔2𝐷g_{1},g_{2}\in D, the map (g1,x)(g2,x)maps-tosubscript𝑔1𝑥subscript𝑔2𝑥(g_{1},x)\mapsto(g_{2},x) distorts the restrictions of ,\langle\cdot,\cdot\rangle to {g1}×Msubscript𝑔1𝑀\{g_{1}\}\times M and {g2}×Msubscript𝑔2𝑀\{g_{2}\}\times M by at most C𝐶C.

The metric then descends to a C1superscript𝐶1C^{1} Riemannian metric on Mαsuperscript𝑀𝛼M^{\alpha}. Note that by averaging the metric over the left action of K𝐾K, we may also assume that the metric on Mαsuperscript𝑀𝛼M^{\alpha} is left-K𝐾K-invariant. This, in particular, implies the right-invariant metric on G𝐺G in (2)2(2) above is chosen to be left-K𝐾K-invariant.

To analyze the coarse dynamics of the suspension action, it is often useful to consider the return cocycle β:G×G/ΓΓ:𝛽𝐺𝐺ΓΓ\beta\colon G\times G/\Gamma\rightarrow\Gamma. This cocycle is defined relative to a fundamental domain \mathcal{F} for the right ΓΓ\Gamma-action on G𝐺G. For any xG/Γ𝑥𝐺Γx\in G/\Gamma, take x~~𝑥\tilde{x} to be the unique lift of x𝑥x in \mathcal{F} and define β(g,x)𝛽𝑔𝑥\beta(g,x) to be the unique element of γΓ𝛾Γ\gamma\in\Gamma such that gx~γ1𝑔~𝑥superscript𝛾1g\tilde{x}\gamma^{-1}\in\mathcal{F}. Any two choices of fundamental domain for ΓΓ\Gamma define cohomologous cocycles but we require a choice of well-controlled fundamental domains \mathcal{F}. Namely, we choose \mathcal{F} to either be contained in a Siegel fundamental set or to be a Dirichlet domain for the identity. With these choices, we have the following.

Let 𝒟~SL(m,)~𝒟SL𝑚\widetilde{\mathcal{D}}\subset\mathrm{SL}(m,\mathbb{R}) denote the Dirichlet domain of the identity for the SL(m,)SL𝑚\mathrm{SL}(m,\mathbb{Z}) action on SL(m,)SL𝑚\mathrm{SL}(m,\mathbb{R}); that is

𝒟~:={gSL(n,):d(g,Id)d(gγ,Id) for all γSL(m,)}.assign~𝒟conditional-set𝑔SL𝑛𝑑𝑔Id𝑑𝑔𝛾Id for all γSL(m,)\widetilde{\mathcal{D}}:=\{g\in\mathrm{SL}(n,\mathbb{R}):d(g,\mathrm{Id})\leq d(g\gamma,\mathrm{Id})\text{ for all $\gamma\in\mathrm{SL}(m,\mathbb{Z})$}\}.

Since each Hi,jsubscript𝐻𝑖𝑗H_{i,j} is geodesically embedded in SL(m,)SL𝑚\mathrm{SL}(m,\mathbb{R}) and since Λi,j=Hi,jSL(m,Z)subscriptΛ𝑖𝑗subscript𝐻𝑖𝑗SL𝑚𝑍\Lambda_{i,j}=H_{i,j}\cap\mathrm{SL}(m,Z), it follows

𝒟:=H1,2𝒟~assign𝒟subscript𝐻12~𝒟\mathcal{D}:=H_{1,2}\cap\widetilde{\mathcal{D}} (7)

is a Dirichlet domain of the identity for the Λ1,2subscriptΛ12\Lambda_{1,2}-action on H1,2subscript𝐻12H_{1,2}. Viewing H1,2SL(2,)similar-to-or-equalssubscript𝐻12SL2H_{1,2}\simeq\mathrm{SL}(2,\mathbb{R}) acting on the upper half-plane model of hyperbolic space 2=SO(2)\SL(2,)superscript2\SO2SL2\mathbb{H}^{2}=\mathrm{SO}(2)\backslash\mathrm{SL}(2,\mathbb{R}) by Möbius transformations SO(2)\𝒟\SO2𝒟{\mathrm{SO}(2)\backslash\mathcal{D}} is the standard Dirichlet domain for the modular surface, the hyperbolic triangle with endpoints at 1/2+i3/212𝑖321/2+i\sqrt{3}/2, 1/2+i3/212𝑖32-1/2+i\sqrt{3}/2, and \infty.

Lemma 2.1.

If \mathcal{F} is either contained in either a Siegel fundamental set or a Dirichlet domain for the identity then there is a constant C𝐶C such that for all gG𝑔𝐺g\in G and xG/Γ𝑥𝐺Γx\in G/\Gamma

(β(g,x))Cd(g,e)+Cd(x,Γ)+C.𝛽𝑔𝑥𝐶𝑑𝑔𝑒𝐶𝑑𝑥Γ𝐶\ell(\beta(g,x))\leq Cd(g,e)+Cd(x,\Gamma)+C.

In the above lemma, \ell is the word-length of β(g,x)𝛽𝑔𝑥\beta(g,x), d(g,e)𝑑𝑔𝑒d(g,e) is the distance from g𝑔g to e𝑒e in G𝐺G, and d(x,Γ)𝑑𝑥Γd(x,\Gamma) is the distance from xG/Γ𝑥𝐺Γx\in G/\Gamma to the identity coset ΓΓ\Gamma in G/Γ𝐺ΓG/\Gamma. For a Dirichlet domain for the identity, the Lemma is shown in [Sha2, §2]; for fundamental domains contained in Siegel fundamental sets, the estimate follows from [FM, Corollary 3.19] and the fact that the distance to the identity in a Siegel domain is quasi-Lipschitz equivalent to the distance to the identity in the quotient G/Γ𝐺ΓG/\Gamma. Both estimates heavily use the main theorem of Lubotzky, Mozes, and Raghunathan [LMR1, LMR2] to compare the word-length of β(g,x)SL(m,)𝛽𝑔𝑥SL𝑚\beta(g,x)\in\mathrm{SL}(m,\mathbb{Z}) with log(β(g,x))norm𝛽𝑔𝑥\log(\|\beta(g,x)\|).

Fix once and for all a fundamental domain 𝒟~SL(m,)~𝒟SL𝑚\mathcal{F}\subset\widetilde{\mathcal{D}}\subset\mathrm{SL}(m,\mathbb{R}).

The estimates in Lemma 2.1 is often used to obtain integrability properties of β𝛽\beta and related cocycles with respect to the Haar measure on G/Γ𝐺ΓG/\Gamma. As the function xd(x,Γ)maps-to𝑥𝑑𝑥Γx\mapsto d(x,\Gamma) is in Lp(G/Γ,Haar)superscript𝐿𝑝𝐺ΓHaarL^{p}(G/\Gamma,\mathrm{Haar}) for any compact set KG𝐾𝐺K\subset G we have that

xsupgK(β(g,x))maps-to𝑥subscriptsupremum𝑔𝐾𝛽𝑔𝑥x\mapsto\sup_{g\in K}\ell(\beta(g,x))

is in Lp(G/Γ,Haar)superscript𝐿𝑝𝐺ΓHaarL^{p}(G/\Gamma,\text{Haar}) for all p1𝑝1p\geq 1. In the sequel, we typically do not directly use the integrability properties (since we work with measures other than Haar) but rather the estimate in Lemma 2.1.

3. Preliminaries on measures, averaging, and Lyapunov exponents

We present a number of technical facts regarding invariant measures, equidistribution, averaging, and Lyapunov exponents that will be used in the remainder of the paper.

3.1. Ratner’s measure classification and equidistribution theorems

We recall Ratner’s theorems on equidistribution of unipotent flows. Let U={u(t)=exp𝔤(tX)}𝑈𝑢𝑡subscript𝔤𝑡𝑋U=\{u(t)=\exp_{\mathfrak{g}}(tX)\} be a 1-parameter unipotent subgroup in G𝐺G. Given any Borel probability measure μ𝜇\mu on G/Γ𝐺ΓG/\Gamma let

UTμ:=1T0Tu(t)μ𝑑t.assignsuperscript𝑈𝑇𝜇1𝑇superscriptsubscript0𝑇𝑢subscript𝑡𝜇differential-d𝑡U^{T}\ast\mu:=\frac{1}{T}\int_{0}^{T}u(t)_{*}\mu\ dt.
Theorem 3.1 (Ratner).

Let U={u(t)=exp𝔤(tX)}𝑈𝑢𝑡subscript𝔤𝑡𝑋U=\{u(t)=\exp_{\mathfrak{g}}(tX)\} be a 1-parameter unipotent subgroup and consider the action on G/Γ𝐺ΓG/\Gamma. The following hold:

  1. (a)

    Every ergodic, U𝑈U-invariant probability measure on G/Γ𝐺ΓG/\Gamma is homogeneous [Rat1, Theorem 1].

  2. (b)

    The orbit closure 𝒪x:={ux:uU}¯assignsubscript𝒪𝑥¯conditional-set𝑢𝑥𝑢𝑈\mathcal{O}_{x}:=\overline{\{u\cdot x:u\in U\}} is homogeneous for every xG/Γ𝑥𝐺Γx\in G/\Gamma [Rat1, Theorem 3].

  3. (c)

    The orbit Ux𝑈𝑥{U\cdot x} equidistributes in 𝒪xsubscript𝒪𝑥\mathcal{O}_{x}; that is UTδxsuperscript𝑈𝑇subscript𝛿𝑥U^{T}\ast\delta_{x} converges to the Haar measure on 𝒪xsubscript𝒪𝑥\mathcal{O}_{x} as T𝑇T\to\infty.

  4. (d)

    Let β𝛽\beta be a root of 𝔤𝔤\mathfrak{g} and let 𝔰𝔩β(2)𝔤𝔰subscript𝔩𝛽2𝔤\mathfrak{sl}_{\beta}(2)\subset\mathfrak{g} be the Lie subalgebra generated by 𝔤βsuperscript𝔤𝛽\mathfrak{g}^{\beta} and 𝔤βsuperscript𝔤𝛽\mathfrak{g}^{-\beta}. Let e,f,h𝔰𝔩β(2)𝑒𝑓𝔰subscript𝔩𝛽2e,f,h\subset\mathfrak{sl}_{\beta}(2) be an 𝔰𝔩(2,)𝔰𝔩2\mathfrak{sl}(2,\mathbb{R}) triple with e𝔤β𝑒superscript𝔤𝛽e\in\mathfrak{g}^{\beta} and f𝔤β𝑓superscript𝔤𝛽f\in\mathfrak{g}^{-\beta} and let 𝔥β=span(h)superscript𝔥𝛽span\mathfrak{h}^{\beta}=\mathrm{span}(h). Let Hβ=exp𝔥βsuperscript𝐻𝛽superscript𝔥𝛽H^{\beta}=\exp\mathfrak{h}^{\beta}.

    Let μ𝜇\mu be a Uβsuperscript𝑈𝛽U^{\beta}-invariant Borel probability measure on G/Γ𝐺ΓG/\Gamma. If μ𝜇\mu is Hβsuperscript𝐻𝛽H^{\beta}-invariant, then μ𝜇\mu is Uβsuperscript𝑈𝛽U^{-\beta}-invariant.

Conclusion (d) follows from [Rat2, Proposition 2.1] and the structure of 𝔰𝔩(2,)𝔰𝔩2\mathfrak{sl}(2,\mathbb{R})-triples. See also the discussion in the paragraph preceding [Rat1, Theorem 9]. In our earlier work on cocompact lattices [BFH], we averaged over higher-dimensional unipotent subgroups and required a variant of (c) due to Nimish Shah [Sha1]. Here we only average over one-dimensional root subgroups and can use the earlier version due to Ratner.

From Theorem 3.1, for any probability measure μ𝜇\mu on G/Γ𝐺ΓG/\Gamma it follows that the weak-* limit

Uμ:=limTUTμassign𝑈𝜇subscript𝑇superscript𝑈𝑇𝜇U\ast\mu:=\lim_{T\to\infty}U^{T}\ast\mu

exists and that the U𝑈U-ergodic components of Uμ𝑈𝜇U\ast\mu are homogeneous.

3.2. Measures with exponentially small mass in the cusps

We now define precisely the notion of measures with exponentially small mass in the cusps from the introduction. Let (X,d)𝑋𝑑(X,d) be a complete, second countable, metric space. Then X𝑋X is Polish. Let μ𝜇\mu be a finite Borel (and hence Radon) measure on X𝑋X. We say that μ𝜇\mu has exponentially small mass in the cusps with exponent ημsubscript𝜂𝜇\eta_{\mu} if for all 0<η<ημ0𝜂subscript𝜂𝜇0<\eta<\eta_{\mu}

Xeηd(x0,x)𝑑μ(x)<subscript𝑋superscript𝑒𝜂𝑑subscript𝑥0𝑥differential-d𝜇𝑥\int_{X}e^{\eta d(x_{0},x)}\ d\mu(x)<\infty (8)

for some (and hence any) choice of base point x0Xsubscript𝑥0𝑋x_{0}\in X. We say that a collection ={μζ}subscript𝜇𝜁\mathcal{M}=\{\mu_{\zeta}\} of probability measures on X𝑋X has uniformly exponentially small mass in the cusps with exponent η0subscript𝜂0\eta_{0} if for all 0<η<η00𝜂subscript𝜂00<\eta<\eta_{0}

supμζ{eηd(x0,x)𝑑μζ(x)}<.subscriptsupremumsubscript𝜇𝜁superscript𝑒𝜂𝑑subscript𝑥0𝑥differential-dsubscript𝜇𝜁𝑥\sup_{\mu_{\zeta}\in\mathcal{M}}\left\{\int e^{\eta d(x_{0},x)}\ d\mu_{\zeta}(x)\right\}<\infty.

Below, we often work in in the setting X=G/Γ𝑋𝐺ΓX=G/\Gamma where G=SL(m,)𝐺SL𝑚G=\mathrm{SL}(m,\mathbb{R}) and Γ=SL(m,)ΓSL𝑚\Gamma=\mathrm{SL}(m,\mathbb{Z}) and where d𝑑d the distance induced from a right-invariant metric on G𝐺G. When X=SL(m,)/SL(m,)𝑋SL𝑚SL𝑚X=\mathrm{SL}(m,\mathbb{R})/\mathrm{SL}(m,\mathbb{Z}) we interpret a point x=gΓG/Γ𝑥𝑔Γ𝐺Γx=g\Gamma\in G/\Gamma as a unimodular lattice Λg=gmsubscriptΛ𝑔𝑔superscript𝑚\Lambda_{g}=g\cdot\mathbb{Z}^{m}. Fix any norm on msuperscript𝑚\mathbb{R}^{m} and define the systole of a lattice ΛmΛsuperscript𝑚\Lambda\subset\mathbb{R}^{m} to be

δ(Λ):=inf{v:vΛ{0}}.\delta(\Lambda):=\inf\left\{\|v\|:v\in\Lambda\smallsetminus\{0\}\right\}.

We have that

c11log(δ(Λg))1+(d(gΓ,eΓ))c2subscript𝑐11𝛿subscriptΛ𝑔1𝑑𝑔Γ𝑒Γsubscript𝑐2c_{1}\leq\frac{1-\log(\delta(\Lambda_{g}))}{1+(d(g\Gamma,e\Gamma))}\leq c_{2} (9)

for some constants whence

C1ec1d(gΓ,eΓ)1δ(Λg)C2ec2d(gΓ,eΓ).subscript𝐶1superscript𝑒subscript𝑐1𝑑𝑔Γ𝑒Γ1𝛿subscriptΛ𝑔subscript𝐶2superscript𝑒subscript𝑐2𝑑𝑔Γ𝑒ΓC_{1}e^{c_{1}d(g\Gamma,e\Gamma)}\leq\frac{1}{\delta(\Lambda_{g})}\leq C_{2}e^{c_{2}d(g\Gamma,e\Gamma)}.

Thus, if we only care about finding a positive exponent ημ>0subscript𝜂𝜇0\eta_{\mu}>0 such that (8) holds for all η<ημ𝜂subscript𝜂𝜇\eta<\eta_{\mu}, it suffices to find η𝜂\eta such that

δ(Λg)η𝑑μ(gΓ)<.𝛿superscriptsubscriptΛ𝑔𝜂differential-d𝜇𝑔Γ\int\delta(\Lambda_{g})^{-\eta}\ d\mu(g\Gamma)<\infty. (10)

We define the systolic exponent ημSsubscriptsuperscript𝜂𝑆𝜇\eta^{S}_{\mu} to be the supremum of all η𝜂\eta satisfying (10).

In the sequel, we will frequently use the following proposition to avoid escape of mass into the cusps of G/Γ𝐺ΓG/\Gamma when averaging a measure along a unipotent flow.

Proposition 3.2.

Let U𝑈U be a 1-parameter unipotent subgroup of G𝐺G. Let μ𝜇\mu be a probability measure on X=SL(m,)/SL(m,)𝑋SL𝑚SL𝑚X=\mathrm{SL}(m,\mathbb{R})/\mathrm{SL}(m,\mathbb{Z}) with exponentially small mass in the cusps. Then the family of measures

{UTμ:T}{Uμ}conditional-setsuperscript𝑈𝑇𝜇𝑇𝑈𝜇\{U^{T}\ast\mu:T\in\mathbb{R}\}\cup\{U\ast\mu\}

has uniformly exponentially small mass in the cusps.

3.3. Proof of Proposition 3.2

We first show that the family of averaged measures

{UTμ:T}conditional-setsuperscript𝑈𝑇𝜇𝑇\{U^{T}\ast\mu:T\in\mathbb{R}\}

has uniformly exponentially small mass in the cusps. The key idea is to use the quantitative non-divergence of unipotent orbits following Kleinbock and Margulis.

Lemma 3.3.

Let μ𝜇\mu be a probability measure on X=SL(m,)/SL(m,)𝑋SL𝑚SL𝑚X=\mathrm{SL}(m,\mathbb{R})/\mathrm{SL}(m,\mathbb{Z}) with exponentially small mass in the cusps and systolic exponent ημSsubscriptsuperscript𝜂𝑆𝜇\eta^{S}_{\mu}.

Then the family of measures {UTμ:T}conditional-setsuperscript𝑈𝑇𝜇𝑇\{U^{T}\ast\mu:T\in\mathbb{R}\} has uniformly exponentially small mass in the cusps with systolic exponent min{ημS,1m2}subscriptsuperscript𝜂𝑆𝜇1superscript𝑚2\min\{\eta^{S}_{\mu},\frac{1}{m^{2}}\}.

Proof.

Let ΔmΔsuperscript𝑚\Delta\subset\mathbb{R}^{m} be a discrete subgroup. Let ΔnormΔ\|\Delta\| denote the volume of Δ/ΔsubscriptΔΔ\Delta_{\mathbb{R}}/\Delta where ΔsubscriptΔ\Delta_{\mathbb{R}} denotes the \mathbb{R}-span of ΔΔ\Delta. It follows from Minkowski’s lemma that there is a constant cmsubscript𝑐𝑚c_{m} (depending only on m𝑚m) such that if

Δ(ρ)rk(Δ)normΔsuperscriptsuperscript𝜌rkΔ\|\Delta\|\leq(\rho^{\prime})^{\mathrm{rk}(\Delta)}

then there is a non-zero vector vΔ𝑣Δv\in\Delta with vcmρnorm𝑣subscript𝑐𝑚superscript𝜌\|v\|\leq c_{m}\rho^{\prime}. In particular, if δ(Λ)ρ𝛿Λ𝜌\delta(\Lambda)\geq\rho then for some constant cmsubscriptsuperscript𝑐𝑚c^{\prime}_{m} we have

Δ(cmρ)rk(Δ)normΔsuperscriptsuperscriptsubscript𝑐𝑚𝜌rkΔ\|\Delta\|\geq(c_{m}^{\prime}\rho)^{\mathrm{rk}(\Delta)}

for all discrete subgroups ΔΛΔΛ\Delta\subset\Lambda.

From [KM, Theorem 5.3] as extended in [Kle, Theorem 0.1], there is a C>1𝐶1C>1 such that for every ΛgG/ΓsubscriptΛ𝑔𝐺Γ\Lambda_{g}\in G/\Gamma and ε>0𝜀0\varepsilon>0, if δ(Λg)ρ𝛿subscriptΛ𝑔𝜌\delta(\Lambda_{g})\geq\rho then, since Δ(cnρ)rk(Δ)normΔsuperscriptsuperscriptsubscript𝑐𝑛𝜌rkΔ\|\Delta\|\geq(c_{n}^{\prime}\rho)^{\mathrm{rk}(\Delta)} for every discrete subgroup ΔΛgΔsubscriptΛ𝑔\Delta\subset\Lambda_{g}, we have

m{t[0,T]:δ(Λutg)ε}C(ε(cn)1ρ)1m2T=C^(ερ)1m2T𝑚conditional-set𝑡0𝑇𝛿subscriptΛsubscript𝑢𝑡𝑔𝜀𝐶superscript𝜀superscriptsuperscriptsubscript𝑐𝑛1𝜌1superscript𝑚2𝑇^𝐶superscript𝜀𝜌1superscript𝑚2𝑇m\{t\in[0,T]:\delta(\Lambda_{u_{t}g})\leq\varepsilon\}\leq C\left(\frac{\varepsilon}{(c_{n}^{\prime})^{-1}\rho}\right)^{\frac{1}{m^{2}}}T=\hat{C}\left(\frac{\varepsilon}{\rho}\right)^{\frac{1}{m^{2}}}T (11)

where m(A)𝑚𝐴m(A) is the Lebesgue measure of the set A𝐴A\subset\mathbb{R}. Note that (11) still holds even in the case ερ𝜀𝜌\varepsilon\geq\rho. Note that if β<1m2𝛽1superscript𝑚2\beta<{\frac{1}{m^{2}}} then for ε<ρ𝜀𝜌\varepsilon<\rho we have

(ερ)1m2T<(ερ)βT.superscript𝜀𝜌1superscript𝑚2𝑇superscript𝜀𝜌𝛽𝑇\left(\frac{\varepsilon}{\rho}\right)^{\frac{1}{m^{2}}}T<\left(\frac{\varepsilon}{\rho}\right)^{\beta}T.

In particular, when β<1m2𝛽1superscript𝑚2\beta<\frac{1}{m^{2}} we have (for all ε>0𝜀0\varepsilon>0 including ε>δ(Λg)𝜀𝛿subscriptΛ𝑔\varepsilon>\delta(\Lambda_{g})) that

m{t[0,T]:δ(Λutg)ε}C^(εδ(Λg))βT.𝑚conditional-set𝑡0𝑇𝛿subscriptΛsubscript𝑢𝑡𝑔𝜀^𝐶superscript𝜀𝛿subscriptΛ𝑔𝛽𝑇m\{t\in[0,T]:\delta(\Lambda_{u_{t}g})\leq\varepsilon\}\leq\hat{C}\left(\frac{\varepsilon}{\delta(\Lambda_{g})}\right)^{\beta}T.

Then for η>0𝜂0\eta>0 and β<1m2𝛽1superscript𝑚2\beta<\frac{1}{m^{2}} we have

[δ(Λg)]η𝑑UTμ(g)superscriptdelimited-[]𝛿subscriptΛ𝑔𝜂differential-dsuperscript𝑈𝑇𝜇𝑔\displaystyle\int[\delta(\Lambda_{g})]^{-\eta}\ dU^{T}\ast\mu(g) =M1T0T[δ(Λutg)]η𝑑t𝑑μ(g)absentsubscript𝑀1𝑇superscriptsubscript0𝑇superscriptdelimited-[]𝛿subscriptΛsubscript𝑢𝑡𝑔𝜂differential-d𝑡differential-d𝜇𝑔\displaystyle=\int_{M}\frac{1}{T}\int_{0}^{T}[\delta(\Lambda_{u_{t}g})]^{-\eta}\ dt\ d\mu(g)
=M1T0m{t[0,T]:[δ(Λutg)]η}𝑑𝑑μ(g)absentsubscript𝑀1𝑇superscriptsubscript0𝑚conditional-set𝑡0𝑇superscriptdelimited-[]𝛿subscriptΛsubscript𝑢𝑡𝑔𝜂differential-ddifferential-d𝜇𝑔\displaystyle=\int_{M}\frac{1}{T}\int_{0}^{\infty}m\{t\in[0,T]:[\delta(\Lambda_{u_{t}g})]^{-\eta}\geq\ell\}\ d\ell\ d\mu(g)
M1T[T+1m{t[0,T]:[δ(Λutg)]η}𝑑]𝑑μ(g)absentsubscript𝑀1𝑇delimited-[]𝑇superscriptsubscript1𝑚conditional-set𝑡0𝑇superscriptdelimited-[]𝛿subscriptΛsubscript𝑢𝑡𝑔𝜂differential-ddifferential-d𝜇𝑔\displaystyle\leq\int_{M}\frac{1}{T}\left[T+\int_{1}^{\infty}m\{t\in[0,T]:[\delta(\Lambda_{u_{t}g})]^{-\eta}\geq\ell\}\ d\ell\right]\ d\mu(g)
=1+M1T1m{t[0,T]:[δ(Λutg)]1η}|ddμ(g)absent1conditionalsubscript𝑀1𝑇superscriptsubscript1𝑚conditional-set𝑡0𝑇delimited-[]𝛿subscriptΛsubscript𝑢𝑡𝑔superscript1𝜂𝑑𝑑𝜇𝑔\displaystyle=1+\int_{M}\frac{1}{T}\int_{1}^{\infty}m\{t\in[0,T]:[\delta(\Lambda_{u_{t}g})]\leq{\ell^{-\frac{1}{\eta}}}\}|\ d\ell\ d\mu(g)
1+M1T1[C^(11ηδ(Λg))βT]𝑑𝑑μ(g)absent1subscript𝑀1𝑇superscriptsubscript1delimited-[]^𝐶superscript1superscript1𝜂𝛿subscriptΛ𝑔𝛽𝑇differential-ddifferential-d𝜇𝑔\displaystyle\leq 1+\int_{M}\frac{1}{T}\int_{1}^{\infty}\left[\hat{C}\left(\frac{1}{\ell^{\frac{1}{\eta}}\delta(\Lambda_{g})}\right)^{\beta}T\right]\ d\ell\ d\mu(g)
=1+C^(M(1δ(Λg))β𝑑μ(g))(1(11η)β𝑑)absent1^𝐶subscript𝑀superscript1𝛿subscriptΛ𝑔𝛽differential-d𝜇𝑔superscriptsubscript1superscript1superscript1𝜂𝛽differential-d\displaystyle=1+\hat{C}\left(\int_{M}\left(\frac{1}{\delta(\Lambda_{g})}\right)^{\beta}\ d\mu(g)\right)\left(\int_{1}^{\infty}\left(\frac{1}{\ell^{\frac{1}{\eta}}}\right)^{\beta}\ d\ell\right)

which is uniformly bounded in T𝑇T as long as η<β<min{ημS,1m2}𝜂𝛽superscriptsubscript𝜂𝜇𝑆1superscript𝑚2\eta<\beta<\min\{\eta_{\mu}^{S},\frac{1}{m^{2}}\}. ∎

For the limit measure Uμ=limTUTμ𝑈𝜇subscript𝑇superscript𝑈𝑇𝜇U\ast\mu=\lim_{T\to\infty}U^{T}\ast\mu we have the following which holds in full generality.

Lemma 3.4.

Let (X,d)𝑋𝑑(X,d) be a complete, second countable, metric space. Let νjsubscript𝜈𝑗\nu_{j} be a sequence of Borel probability measures on X𝑋X converging in the weak-* topology to a measure ν𝜈\nu. If the family {νj}subscript𝜈𝑗\{\nu_{j}\} has uniformly exponentially small mass in the cusps with exponent η0subscript𝜂0\eta_{0} then the limit ν𝜈\nu has exponentially small mass in the cusps with exponent η0subscript𝜂0\eta_{0}.

Proof.

We have that νjνsubscript𝜈𝑗𝜈\nu_{j}\to\nu in the weak-* topology. In particular, for any closed set CX𝐶𝑋C\subset X and open set UX𝑈𝑋U\subset X we have

lim supjνj(C)ν(C)andlim infjνj(U)ν(U).formulae-sequencesubscriptlimit-supremum𝑗subscript𝜈𝑗𝐶𝜈𝐶andsubscriptlimit-infimum𝑗subscript𝜈𝑗𝑈𝜈𝑈\limsup_{j\to\infty}\nu_{j}(C)\leq\nu(C)\quad\text{and}\quad\liminf_{j\to\infty}\nu_{j}(U)\geq\nu(U).

Fix 0<η<η<η00superscript𝜂𝜂subscript𝜂00<\eta^{\prime}<\eta<\eta_{0} and take δ:=ηη1.assign𝛿𝜂superscript𝜂1\delta:=\frac{\eta}{\eta^{\prime}}-1. Fix N𝑁N with

eηd(x,x0)𝑑νj(x)<Nsuperscript𝑒𝜂𝑑𝑥subscript𝑥0differential-dsubscript𝜈𝑗𝑥𝑁\int e^{\eta d(x,x_{0})}\ d\nu_{j}(x)<N

for all j𝑗j. Using Markov’s inequality, for all M>0𝑀0M>0 and every j𝑗j we have

νj{x:eηd(x,x0)>M}N/Msubscript𝜈𝑗conditional-set𝑥superscript𝑒𝜂𝑑𝑥subscript𝑥0𝑀𝑁𝑀\nu_{j}\{x:e^{\eta d(x,x_{0})}>M\}\leq N/M

so

ν{x:eηd(x0,x)>M}N/M.𝜈conditional-set𝑥superscript𝑒𝜂𝑑subscript𝑥0𝑥𝑀𝑁𝑀\nu\{x:e^{\eta d(x_{0},x)}>M\}\leq N/M.

Then, for the limit measure ν𝜈\nu, we have

G/Γeηd(x0,x)𝑑ν(x)subscript𝐺Γsuperscript𝑒superscript𝜂𝑑subscript𝑥0𝑥differential-d𝜈𝑥\displaystyle\int_{G/\Gamma}e^{\eta^{\prime}d(x_{0},x)}\ d\nu(x) =0ν{x:eηd(x0,x)M}𝑑Mabsentsuperscriptsubscript0𝜈conditional-set𝑥superscript𝑒superscript𝜂𝑑subscript𝑥0𝑥𝑀differential-d𝑀\displaystyle=\int_{0}^{\infty}\nu\{x:e^{\eta^{\prime}d(x_{0},x)}\geq M\}\ dM
=0ν{x:(eηd(x0,x))1/(1+δ)M}𝑑Mabsentsuperscriptsubscript0𝜈conditional-set𝑥superscriptsuperscript𝑒𝜂𝑑subscript𝑥0𝑥11𝛿𝑀differential-d𝑀\displaystyle=\int_{0}^{\infty}\nu\{x:\left(e^{\eta d(x_{0},x)}\right)^{1/(1+\delta)}\geq{M}\}\ dM
=0ν{x:eηd(x0,x)M1+δ}𝑑Mabsentsuperscriptsubscript0𝜈conditional-set𝑥superscript𝑒𝜂𝑑subscript𝑥0𝑥superscript𝑀1𝛿differential-d𝑀\displaystyle=\int_{0}^{\infty}\nu\{x:e^{\eta d(x_{0},x)}\geq{M}^{1+\delta}\}\ dM
1+1NM1+δ𝑑M.absent1superscriptsubscript1𝑁superscript𝑀1𝛿differential-d𝑀\displaystyle\leq 1+\int_{1}^{\infty}\frac{N}{{M}^{1+\delta}}\ dM.\qed

3.4. Averaging certain measures on SL(m,)/SL(m,)SL𝑚SL𝑚\mathrm{SL}(m,\mathbb{R})/\mathrm{SL}(m,\mathbb{Z})

Take {α1,,αm}subscript𝛼1subscript𝛼𝑚\{\alpha_{1},\dots,\alpha_{m}\} to be the standard set of simple positive roots of SL(m,)SL𝑚\mathrm{SL}(m,\mathbb{R}):

αj(diag(et1,,etm))=tjtj+1.subscript𝛼𝑗diagsuperscript𝑒subscript𝑡1superscript𝑒subscript𝑡𝑚subscript𝑡𝑗subscript𝑡𝑗1\alpha_{j}(\mathrm{diag}(e^{t_{1}},\dots,e^{t_{m}}))=t_{j}-t_{j+1}.

Let H1subscript𝐻1H_{1} be the analytic subgroup of SL(m,)SL𝑚\mathrm{SL}(m,\mathbb{R}) whose Lie algebra is generated by roots spaces associated to {±α1}plus-or-minussubscript𝛼1\{\pm\alpha_{1}\} and let H2subscript𝐻2H_{2} be the analytic subgroup of SL(m,)SL𝑚\mathrm{SL}(m,\mathbb{R}) whose Lie algebra is generated by roots spaces associated to {±α3,,±αn}plus-or-minussubscript𝛼3plus-or-minussubscript𝛼𝑛\{\pm\alpha_{3},\dots,\pm\alpha_{n}\}. We have H1SL(2,)subscript𝐻1SL2H_{1}\cong\mathrm{SL}(2,\mathbb{R}) and H2SL(m2,)subscript𝐻2SL𝑚2H_{2}\cong\mathrm{SL}(m-2,\mathbb{R}). Then H=H1×H2SL(m,)𝐻subscript𝐻1subscript𝐻2SL𝑚H=H_{1}\times H_{2}\subset\mathrm{SL}(m,\mathbb{R}) is the subgroup of all matrices of the form

(B00C)𝐵00𝐶\left(\begin{array}[]{cc}B&0\\ 0&C\end{array}\right)

where det(B)=det(C)=1𝐵𝐶1\det(B)=\det(C)=1.

We let Asuperscript𝐴A^{\prime} be the the co-rank-1 subgroup AAsuperscript𝐴𝐴A^{\prime}\subset A of the Cartan subgroup A𝐴A given by A=AHsuperscript𝐴𝐴𝐻A^{\prime}=A\cap H. Let δ=α1++αn𝛿subscript𝛼1subscript𝛼𝑛\delta=\alpha_{1}+\dots+\alpha_{n} be the highest positive root.

Proposition 3.5.

Let μ𝜇\mu be any H𝐻H-invariant probability on SL(m,)/SL(m,)SL𝑚SL𝑚\mathrm{SL}(m,\mathbb{R})/\mathrm{SL}(m,\mathbb{Z}). Let β=α2superscript𝛽subscript𝛼2\beta^{\prime}=\alpha_{2} or β=δsuperscript𝛽𝛿\beta^{\prime}=\delta and let β^=α2^𝛽subscript𝛼2\hat{\beta}=-\alpha_{2} or β^=δ^𝛽𝛿\hat{\beta}=-\delta.

Then Uβμsuperscript𝑈superscript𝛽𝜇U^{\beta^{\prime}}\ast\mu is H𝐻H-invariant and

Uβ^Uβμsuperscript𝑈^𝛽superscript𝑈superscript𝛽𝜇U^{\hat{\beta}}\ast U^{\beta^{\prime}}\ast\mu

is the Haar measure on G/Γ𝐺ΓG/\Gamma.

Proof.

We have that μ𝜇\mu is Asuperscript𝐴A^{\prime}-invariant. Let μ=Uβμsuperscript𝜇superscript𝑈superscript𝛽𝜇\mu^{\prime}=U^{\beta^{\prime}}\ast\mu and note that μsuperscript𝜇\mu^{\prime} remains H𝐻H- and Asuperscript𝐴A^{\prime}-invariant.

Case 1(a) : β=α2superscript𝛽subscript𝛼2\beta^{\prime}=\alpha_{2}. Consider first the case that β=α2superscript𝛽subscript𝛼2\beta^{\prime}=\alpha_{2}. Then μsuperscript𝜇\mu^{\prime} remains invariant under Uα1superscript𝑈subscript𝛼1U^{-\alpha_{1}} and Uαjsuperscript𝑈subscript𝛼𝑗U^{-\alpha_{j}} for all 3jn3𝑗𝑛3\leq j\leq n since these roots commute with βsuperscript𝛽\beta^{\prime}. By Theorem 3.1(d) we have that μsuperscript𝜇\mu^{\prime} is also invariant under Uα1superscript𝑈subscript𝛼1U^{\alpha_{1}} and Uαjsuperscript𝑈subscript𝛼𝑗U^{\alpha_{j}} for all 3jn3𝑗𝑛3\leq j\leq n. Taking brackets, μsuperscript𝜇\mu^{\prime} is invariant under Uβsuperscript𝑈𝛽U^{\beta} for every positive root βΣ+𝛽subscriptΣ\beta\in\Sigma_{+}.

Case 1(b) : β=δsuperscript𝛽𝛿\beta^{\prime}=\delta. Consider now the case that β=δsuperscript𝛽𝛿\beta^{\prime}=\delta. Then μsuperscript𝜇\mu^{\prime} remains invariant under Uα1superscript𝑈subscript𝛼1U^{\alpha_{1}} and Uαjsuperscript𝑈subscript𝛼𝑗U^{\alpha_{j}} for all 3jn3𝑗𝑛3\leq j\leq n since these roots commute with δ𝛿\delta. By Theorem 3.1(d) we have that μsuperscript𝜇\mu^{\prime} is also invariant under Uα1superscript𝑈subscript𝛼1U^{-\alpha_{1}} and Uαjsuperscript𝑈subscript𝛼𝑗U^{-\alpha_{j}} for all 3jn3𝑗𝑛3\leq j\leq n. Taking brackets, μsuperscript𝜇\mu^{\prime} is invariant under Uβsuperscript𝑈𝛽U^{\beta} for every positive root β𝛽\beta of the form δαnαn1αj=α1++αj1𝛿subscript𝛼𝑛subscript𝛼𝑛1subscript𝛼𝑗subscript𝛼1subscript𝛼𝑗1\delta-\alpha_{n}-\alpha_{n-1}-\dots-\alpha_{j}=\alpha_{1}+\dots+\alpha_{j-1} for each j3𝑗3j\geq 3. In particular, μsuperscript𝜇\mu^{\prime} is invariant under Uα1+α2superscript𝑈subscript𝛼1subscript𝛼2U^{\alpha_{1}+\alpha_{2}} and hence also invariant under Uα2superscript𝑈subscript𝛼2U^{\alpha_{2}}. In particular μsuperscript𝜇\mu^{\prime} is invariant under Uβsuperscript𝑈𝛽U^{\beta} for every positive root βΣ+𝛽subscriptΣ\beta\in\Sigma_{+}.

Note that in either case, we have that μsuperscript𝜇\mu^{\prime} is invariant under Uβsuperscript𝑈𝛽U^{\beta} for every positive root βΣ+𝛽subscriptΣ\beta\in\Sigma_{+}.

Let μ^=Uβ^μ^𝜇superscript𝑈^𝛽superscript𝜇\hat{\mu}=U^{\hat{\beta}}\ast\mu^{\prime}.

Case 2(a) : β^=α2^𝛽subscript𝛼2\hat{\beta}=-\alpha_{2}. If β^=α2^𝛽subscript𝛼2\hat{\beta}=-\alpha_{2}, then μ^^𝜇\hat{\mu} remains invariant under Uα1superscript𝑈subscript𝛼1U^{\alpha_{1}} and Uαjsuperscript𝑈subscript𝛼𝑗U^{\alpha_{j}} for all 3jn3𝑗𝑛3\leq j\leq n. Note additionally μ^^𝜇\hat{\mu} remains invariant under the highest-root group Uδsuperscript𝑈𝛿U^{\delta}. Again, by Theorem 3.1(d) we have that μ^^𝜇\hat{\mu} is also invariant under Uα1superscript𝑈subscript𝛼1U^{-\alpha_{1}} and Uαjsuperscript𝑈subscript𝛼𝑗U^{-\alpha_{j}} for all 3jn3𝑗𝑛3\leq j\leq n. In particular μ^^𝜇\hat{\mu} is also invariant under Uβsuperscript𝑈𝛽U^{\beta} for every negative root βΣ𝛽subscriptΣ\beta\in\Sigma_{-}. It follows as in Case 1(b) that μ^^𝜇\hat{\mu} is invariant under Uα2superscript𝑈subscript𝛼2U^{\alpha_{2}} and hence invariant under Uβsuperscript𝑈𝛽U^{\beta} for every positive root βΣ+𝛽subscriptΣ\beta\in\Sigma_{+}. Thus μ𝜇\mu is G𝐺G-invariant.

Case 2(b) : β^=δ^𝛽𝛿\hat{\beta}=-\delta. If β^=δ^𝛽𝛿\hat{\beta}=-\delta, then μ^^𝜇\hat{\mu} remains invariant under Uα1superscript𝑈subscript𝛼1U^{-\alpha_{1}} and Uαjsuperscript𝑈subscript𝛼𝑗U^{-\alpha_{j}} for all 3jn3𝑗𝑛3\leq j\leq n. Note additionally μ^^𝜇\hat{\mu} remains invariant under Uα2superscript𝑈subscript𝛼2U^{\alpha_{2}}. Again, we have that μ^^𝜇\hat{\mu} is also invariant under Uα1superscript𝑈subscript𝛼1U^{\alpha_{1}} and Uαjsuperscript𝑈subscript𝛼𝑗U^{\alpha_{j}} for all 3jn3𝑗𝑛3\leq j\leq n. In particular μ^^𝜇\hat{\mu} is also invariant under Uβsuperscript𝑈𝛽U^{\beta} for every positive root βΣ+𝛽subscriptΣ\beta\in\Sigma_{+}. As in Case 1(b) that μ^^𝜇\hat{\mu} is invariant under Uα2superscript𝑈subscript𝛼2U^{-\alpha_{2}} and hence invariant under Uβsuperscript𝑈𝛽U^{\beta} for every negative root βΣ𝛽subscriptΣ\beta\in\Sigma_{-}. Thus μ𝜇\mu is G𝐺G-invariant. ∎

3.5. Lyapunov exponents for unbounded cocycles

Let (X,d)𝑋𝑑(X,d) be a second countable, complete metric space. We moreover assume the metric d𝑑d is proper. Let G𝐺G act continuously on X𝑋X.

Let X𝑋\mathcal{E}\to X be a continuous, finite-dimensional vector bundle equipped with a norm \|\cdot\|. A linear cocycle over the G𝐺G-action on X𝑋X is an action 𝒜:G×:𝒜𝐺\mathcal{A}\colon G\times\mathcal{E}\to\mathcal{E} by vector-bundle automorphisms that projects to the G𝐺G-action on X𝑋X. We write 𝒜(g,x)𝒜𝑔𝑥\mathcal{A}(g,x) for the linear map between Banach spaces xsubscript𝑥\mathcal{E}_{x} and gxsubscript𝑔𝑥\mathcal{E}_{g\cdot x}. By the norm of 𝒜(g,x)𝒜𝑔𝑥\mathcal{A}(g,x) we mean the operator norm and the conorm is m(𝒜(g,x))=𝒜(g,x)11𝑚𝒜𝑔𝑥superscriptnorm𝒜superscript𝑔𝑥11m(\mathcal{A}(g,x))=\|\mathcal{A}(g,x)^{-1}\|^{-1}. We say that 𝒜𝒜\mathcal{A} is tempered with respect to the metric d𝑑d if there is a k0𝑘0k\geq 0 such that for any compact set KG𝐾𝐺K\subset G and base point x0Xsubscript𝑥0𝑋x_{0}\in X there is C>1𝐶1C>1 so that

supgK𝒜(g,x)Cekd(x,x0)subscriptsupremum𝑔𝐾norm𝒜𝑔𝑥𝐶superscript𝑒𝑘𝑑𝑥subscript𝑥0\sup_{g\in K}\|\mathcal{A}(g,x)\|\leq Ce^{kd(x,x_{0})}

and

infgKm(𝒜(g,x))1Cekd(x,x0)subscriptinfimum𝑔𝐾𝑚𝒜𝑔𝑥1𝐶superscript𝑒𝑘𝑑𝑥subscript𝑥0\inf_{g\in K}m(\mathcal{A}(g,x))\geq\frac{1}{C}e^{-kd(x,x_{0})}

where \|\cdot\| denotes the operator norm and m()𝑚m(\cdot) denotes the operator conorm applied to linear maps between Banach spaces xsubscript𝑥\mathcal{E}_{x} and gxsubscript𝑔𝑥\mathcal{E}_{g\cdot x}.

If μ𝜇\mu is a probability measure on (X,d)𝑋𝑑(X,d) with exponentially small mass in the cusps, it follows that the function xd(x,x0)maps-to𝑥𝑑𝑥subscript𝑥0x\mapsto d(x,x_{0}) is L1(μ)superscript𝐿1𝜇L^{1}(\mu) whence we immediately obtain the following.

Claim 3.6.

Let μ𝜇\mu a probability measure on X𝑋X with exponentially small mass in the cusps. Suppose that 𝒜𝒜\mathcal{A} is tempered. Then for any compact KG𝐾𝐺K\subset G, the functions

xsupsKlog𝒜(s,x),xinfsKlogm(𝒜(s,x))formulae-sequencemaps-to𝑥subscriptsupremum𝑠𝐾norm𝒜𝑠𝑥maps-to𝑥subscriptinfimum𝑠𝐾𝑚𝒜𝑠𝑥x\mapsto\sup_{s\in K}\log\left\|\mathcal{A}(s,x)\right\|,\quad\quad x\mapsto\inf_{s\in K}\log m\left(\mathcal{A}(s,x)\right)

are L1(μ)superscript𝐿1𝜇L^{1}(\mu).

Given sG𝑠𝐺s\in G and an s𝑠s-invariant Borel probability measure μ𝜇\mu on X𝑋X we define the average leading (or top) Lyapunov exponent of 𝒜𝒜\mathcal{A} to be

λtop,s,μ,𝒜:=infn1nlog𝒜(sn,x)dμ(x).assignsubscript𝜆top𝑠𝜇𝒜subscriptinfimum𝑛1𝑛norm𝒜superscript𝑠𝑛𝑥𝑑𝜇𝑥\lambda_{\mathrm{top},s,\mu,\mathcal{A}}:=\inf_{n\to\infty}\frac{1}{n}\int\log\|\mathcal{A}(s^{n},x)\|\ d\mu(x). (12)

From the integrability of the function xlog𝒜(s,x)maps-to𝑥norm𝒜𝑠𝑥x\mapsto\log\|\mathcal{A}(s,x)\| we obtain the finiteness of Lyapunov exponents.

Corollary 3.7.

For sG𝑠𝐺s\in G and μ𝜇\mu an s𝑠s-invariant probability measure on X𝑋X with exponentially small mass in the cusps, if 𝒜𝒜\mathcal{A} is tempered then the average leading Lyapunov exponent λtop,s,μ,𝒜subscript𝜆top𝑠𝜇𝒜\lambda_{\mathrm{top},s,\mu,\mathcal{A}} of 𝒜𝒜\mathcal{A} is finite.

Note that for an s𝑠s-invariant measure μ𝜇\mu, the sequence log𝒜(sn,x)dμ(x)norm𝒜superscript𝑠𝑛𝑥𝑑𝜇𝑥\int\log\|\mathcal{A}(s^{n},x)\|\ d\mu(x) is subadditive whence the infimum in (12) maybe replaced by a limit.

As in the case of bounded continuous linear cocycles, we obtain upper-semicontinuity of leading Lyapunov exponents for continuous tempered cocycles when restricted to families of measures with uniformly exponentially small measure in the cusp.

Lemma 3.8.

Let 𝒜𝒜\mathcal{A} be a tempered cocycle. Given sG𝑠𝐺s\in G suppose the restriction of the cocycle 𝒜:G×:𝒜𝐺\mathcal{A}\colon G\times\mathcal{E}\to\mathcal{E} to the action of s𝑠s is continuous.

Then—when restricted to a set of s𝑠s-invariant Borel probability measures with uniformly exponentially small mass in the cusps—the function

μλtop,s,μ,𝒜maps-to𝜇subscript𝜆top𝑠𝜇𝒜\mu\mapsto\lambda_{\mathrm{top},s,\mu,\mathcal{A}}

is upper-semicontinuous with respect to the weak-* topology.

Proof.

Let ={μζ}ζsubscriptsubscript𝜇𝜁𝜁\mathcal{M}=\{\mu_{\zeta}\}_{\zeta\in\mathcal{I}} be a family of s𝑠s-invariant Borel probability measures with uniformly exponentially small mass in the cusps. As the pointwise infimum of continuous functions is upper-semicontinuous, is enough to show that the function

,μlog𝒜(sn,x)dμ(x)formulae-sequencemaps-to𝜇norm𝒜superscript𝑠𝑛𝑥𝑑𝜇𝑥\mathcal{M}\to\mathbb{R},\quad\quad\mu\mapsto\int\log\|\mathcal{A}(s^{n},x)\|\ d\mu(x)

is continuous with respect to the weak-* topology for each n𝑛n. As the weak-* topology is first countable, it is enough to show μlog𝒜(sn,x)dμ(x)maps-to𝜇norm𝒜superscript𝑠𝑛𝑥𝑑𝜇𝑥\mu\mapsto\int\log\|\mathcal{A}(s^{n},x)\|\ d\mu(x) is sequentially continuous.

Let μjμsubscript𝜇𝑗subscript𝜇\mu_{j}\to\mu_{\infty} in \mathcal{M}. Given M>0𝑀0M>0, fix a continuous ψM:X[0,1]:subscript𝜓𝑀𝑋01\psi_{M}\colon X\to[0,1] with

ψM(x)=1 if d(x,x0)M and ψM(x)=0 if d(x,x0)M+1.ψM(x)=1 if d(x,x0)M and ψM(x)=0 if d(x,x0)M+1\text{$\psi_{M}(x)=1$ if $d(x,x_{0})\leq M$ and $\psi_{M}(x)=0$ if $d(x,x_{0})\geq M+1$}.

As we assume our metric is proper, xψM(x)log𝒜(sn,x)maps-to𝑥subscript𝜓𝑀𝑥norm𝒜superscript𝑠𝑛𝑥x\mapsto\psi_{M}(x)\log\|\mathcal{A}(s^{n},x)\| is a bounded continuous function whence

logψM(x)log𝒜(sn,x)dμj(x)ψM(x)log𝒜(sn,x)dμ(x).subscript𝜓𝑀𝑥norm𝒜superscript𝑠𝑛𝑥𝑑subscript𝜇𝑗𝑥subscript𝜓𝑀𝑥norm𝒜superscript𝑠𝑛𝑥𝑑subscript𝜇𝑥\int\log\psi_{M}(x)\log\|\mathcal{A}(s^{n},x)\|\ d\mu_{j}(x)\to\int\psi_{M}(x)\log\|\mathcal{A}(s^{n},x)\|\ d\mu_{\infty}(x).

Moreover, there are C>1,k1,formulae-sequence𝐶1𝑘1C>1,k\geq 1, and η>0𝜂0\eta>0 such that for all xX𝑥𝑋x\in X and μζsubscript𝜇𝜁\mu_{\zeta}\in\mathcal{M}

logCkd(x,x0)log𝒜(sn,x)logC+kd(x,x0),𝐶𝑘𝑑𝑥subscript𝑥0norm𝒜superscript𝑠𝑛𝑥𝐶𝑘𝑑𝑥subscript𝑥0-\log C-{kd(x,x_{0})}\leq\log\|\mathcal{A}(s^{n},x)\|\leq\log C+{kd(x,x_{0})},

and

eηd(x,x0)𝑑μζ(x)C.superscript𝑒𝜂𝑑𝑥subscript𝑥0differential-dsubscript𝜇𝜁𝑥𝐶\int e^{\eta d(x,x_{0})}\ d\mu_{\zeta}(x)\leq C.

In particular,

μζ({x:d(x,x0)M})CeηM.subscript𝜇𝜁conditional-set𝑥𝑑𝑥subscript𝑥0𝑀𝐶superscript𝑒𝜂𝑀\mu_{\zeta}(\{x:d(x,x_{0})\geq M\})\leq Ce^{-\eta M}.

Thus for any μζsubscript𝜇𝜁\mu_{\zeta}\in\mathcal{M}, we have

|log𝒜(sn,x)conditionalnorm𝒜superscript𝑠𝑛𝑥\displaystyle\int\big{|}\log\|\mathcal{A}(s^{n},x)\| ψM(x)log𝒜(sn,x)|dμζ(x)conditionalsubscript𝜓𝑀𝑥norm𝒜superscript𝑠𝑛𝑥𝑑subscript𝜇𝜁𝑥\displaystyle-\psi_{M}(x)\log\|\mathcal{A}(s^{n},x)\|\big{|}\ d\mu_{\zeta}(x)
{x:d(x,x0)M}|log𝒜(sn,x)ψM(x)log𝒜(sn,x)|𝑑μζ(x)absentsubscriptconditional-set𝑥𝑑𝑥subscript𝑥0𝑀norm𝒜superscript𝑠𝑛𝑥subscript𝜓𝑀𝑥norm𝒜superscript𝑠𝑛𝑥differential-dsubscript𝜇𝜁𝑥\displaystyle\leq\int_{\{x:d(x,x_{0})\geq M\}}\big{|}\log\|\mathcal{A}(s^{n},x)\|-\psi_{M}(x)\log\|\mathcal{A}(s^{n},x)\|\big{|}\ d\mu_{\zeta}(x)
{x:d(x,x0)M}|log𝒜(sn,x)|𝑑μζ(x)absentsubscriptconditional-set𝑥𝑑𝑥subscript𝑥0𝑀norm𝒜superscript𝑠𝑛𝑥differential-dsubscript𝜇𝜁𝑥\displaystyle\leq\int_{\{x:d(x,x_{0})\geq M\}}\big{|}\log\|\mathcal{A}(s^{n},x)\|\big{|}\ d\mu_{\zeta}(x)
{x:d(x,x0)M}logC+kd(x,x0)dμζ(x)absentsubscriptconditional-set𝑥𝑑𝑥subscript𝑥0𝑀𝐶𝑘𝑑𝑥subscript𝑥0𝑑subscript𝜇𝜁𝑥\displaystyle\leq\int_{\{x:d(x,x_{0})\geq M\}}\log C+{kd(x,x_{0})}\ d\mu_{\zeta}(x)
(logC)CeηM+k{x:d(x,x0)M}d(x,x0)𝑑μζ(x)absent𝐶𝐶superscript𝑒𝜂𝑀𝑘subscriptconditional-set𝑥𝑑𝑥subscript𝑥0𝑀𝑑𝑥subscript𝑥0differential-dsubscript𝜇𝜁𝑥\displaystyle\leq(\log C)Ce^{-\eta M}+k\int_{\{x:d(x,x_{0})\geq M\}}{d(x,x_{0})}\ d\mu_{\zeta}(x)
(logC+kM)CeηM+k=Mμζ{x:d(x,x0)}𝑑absent𝐶𝑘𝑀𝐶superscript𝑒𝜂𝑀𝑘superscriptsubscript𝑀subscript𝜇𝜁conditional-set𝑥𝑑𝑥subscript𝑥0differential-d\displaystyle\leq(\log C+kM)Ce^{-\eta M}+k\int_{\ell=M}^{\infty}\mu_{\zeta}\{x:{d(x,x_{0})}\geq\ell\}\ d\ell
(logC+kM)CeηM+k=MCeη𝑑absent𝐶𝑘𝑀𝐶superscript𝑒𝜂𝑀𝑘superscriptsubscript𝑀𝐶superscript𝑒𝜂differential-d\displaystyle\leq(\log C+kM)Ce^{-\eta M}+k\int_{\ell=M}^{\infty}Ce^{-\eta\ell}\ d\ell
(logC+kM)CeηM+kCeη(M)η.absent𝐶𝑘𝑀𝐶superscript𝑒𝜂𝑀𝑘𝐶superscript𝑒𝜂𝑀𝜂\displaystyle\leq(\log C+kM)Ce^{-\eta M}+k\frac{Ce^{\eta(-M)}}{\eta}.

It follows that given ε>0𝜀0\varepsilon>0 there is M𝑀M so that

|log𝒜(sn,x)ψM(x)log𝒜(sn,x)|𝑑μζ(x)εnorm𝒜superscript𝑠𝑛𝑥subscript𝜓𝑀𝑥norm𝒜superscript𝑠𝑛𝑥differential-dsubscript𝜇𝜁𝑥𝜀\int\big{|}\log\|\mathcal{A}(s^{n},x)\|-\psi_{M}(x)\log\|\mathcal{A}(s^{n},x)\|\big{|}\ d\mu_{\zeta}(x)\\ \leq\varepsilon

for all μζsubscript𝜇𝜁\mu_{\zeta}\in\mathcal{M}.

In particular, taking M𝑀M and j𝑗j sufficiently large we have

|\displaystyle\Big{|}\int log𝒜(sn,)dμlog𝒜(sn,)dμj|\displaystyle\log\|\mathcal{A}(s^{n},\cdot)\|\ d\mu_{\infty}-\int\log\|\mathcal{A}(s^{n},\cdot)\|\ d\mu_{j}\Big{|}
|log𝒜(sn,)ψMlog𝒜(sn,)|𝑑μabsentnorm𝒜superscript𝑠𝑛subscript𝜓𝑀norm𝒜superscript𝑠𝑛differential-dsubscript𝜇\displaystyle\leq\int\big{|}\log\|\mathcal{A}(s^{n},\cdot)\|-\psi_{M}\log\|\mathcal{A}(s^{n},\cdot)\|\big{|}\ d\mu_{\infty}
+|ψMlog𝒜(sn,)dμjψMlog𝒜(sn,)dμ|delimited-|‖subscript𝜓𝑀𝒜superscript𝑠𝑛norm𝑑subscript𝜇𝑗subscript𝜓𝑀𝒜superscript𝑠𝑛delimited-‖|𝑑subscript𝜇\displaystyle\quad\quad+\Big{|}\int\psi_{M}\log\|\mathcal{A}(s^{n},\cdot)\|\ d\mu_{j}-\int\psi_{M}\log\|\mathcal{A}(s^{n},\cdot)\|\ d\mu_{\infty}\Big{|}
+|log𝒜(sn,)ψMlog𝒜(sn,)|𝑑μjnorm𝒜superscript𝑠𝑛subscript𝜓𝑀norm𝒜superscript𝑠𝑛differential-dsubscript𝜇𝑗\displaystyle\quad\quad+\int\big{|}\log\|\mathcal{A}(s^{n},\cdot)\|-\psi_{M}\log\|\mathcal{A}(s^{n},\cdot)\|\big{|}\ d\mu_{j}
3ε.absent3𝜀\displaystyle\leq 3\varepsilon.

Sequential continuity then follows. ∎

3.6. Lyapunov exponents under averaging and limits

We now consider the behavior of the top Lyapunov exponent λtop,s,μ,𝒜subscript𝜆top𝑠𝜇𝒜\lambda_{\mathrm{top},s,\mu,\mathcal{A}} as we average an s𝑠s-invariant probability measure μ𝜇\mu over an amenable subgroup of G𝐺G contained in the centralizer of s𝑠s.

Lemma 3.9.

Let sG𝑠𝐺s\in G and let μ𝜇\mu be an s𝑠s-invariant probability measure on X𝑋X with exponentially small mass in the cusps. Let 𝒜:G×:𝒜𝐺\mathcal{A}\colon G\times\mathcal{E}\to\mathcal{E} be a tempered continuous cocycle.

For any amenable subgroup HCG(s)𝐻subscript𝐶𝐺𝑠H\subset C_{G}(s) and any Følner sequence of precompact sets Fnsubscript𝐹𝑛F_{n} in H𝐻H, if the family {Fnμ}subscript𝐹𝑛𝜇\{F_{n}\ast\mu\} has uniformly exponentially small mass in the cusps then for any subsequential limit μsuperscript𝜇\mu^{\prime} of {Fnμ}subscript𝐹𝑛𝜇\{F_{n}\ast\mu\} we have

λtop,s,μ,𝒜λtop,s,μ,𝒜.subscript𝜆top𝑠𝜇𝒜subscript𝜆top𝑠superscript𝜇𝒜\lambda_{\mathrm{top},s,\mu,\mathcal{A}}\leq\lambda_{\mathrm{top},s,\mu^{\prime},\mathcal{A}}.
Proof.

First note that Lemma 3.4 implies the family {Fnμ}{μ}subscript𝐹𝑛𝜇superscript𝜇\{F_{n}\ast\mu\}\cup\{\mu^{\prime}\} has uniformly exponentially small mass in the cusps. Note also that for every m𝑚m, the measure Fmμsubscript𝐹𝑚𝜇F_{m}\ast\mu is s𝑠s-invariant.

We first claim that λtop,s,Fmμ,𝒜=λtop,s,μ,𝒜subscript𝜆top𝑠subscript𝐹𝑚𝜇𝒜subscript𝜆top𝑠𝜇𝒜\lambda_{\mathrm{top},s,F_{m}\ast\mu,\mathcal{A}}=\lambda_{\mathrm{top},s,\mu,\mathcal{A}} for every m𝑚m. For tH𝑡𝐻t\in H define ct(x)=sup{𝒜(t,x),m(𝒜(t,x))1}subscript𝑐𝑡𝑥supremumnorm𝒜𝑡𝑥𝑚superscript𝒜𝑡𝑥1c_{t}(x)=\sup\{\|\mathcal{A}(t,x)\|,m(\mathcal{A}(t,x))^{-1}\} and let cm(x)=suptFmct(x)subscript𝑐𝑚𝑥subscriptsupremum𝑡subscript𝐹𝑚subscript𝑐𝑡𝑥c_{m}(x)=\sup_{t\in F_{m}}c_{t}(x). As Fmsubscript𝐹𝑚F_{m} is precompact, from Claim 3.6 we have that logcmL1(μ)subscript𝑐𝑚superscript𝐿1𝜇\log c_{m}\in L^{1}(\mu).

For xM𝑥𝑀x\in M and tFm𝑡subscript𝐹𝑚t\in F_{m}, the cocycle property and subadditivity of norms yields

log𝒜(sn,tx)norm𝒜superscript𝑠𝑛𝑡𝑥\displaystyle\log\|\mathcal{A}(s^{n},tx)\| log𝒜(t1,tx)+log𝒜(sn,x)+log𝒜(t,snx)absentnorm𝒜superscript𝑡1𝑡𝑥norm𝒜superscript𝑠𝑛𝑥norm𝒜𝑡superscript𝑠𝑛𝑥\displaystyle\leq\log\|\mathcal{A}(t^{-1},tx)\|+\log\|\mathcal{A}(s^{n},x)\|+\log\|\mathcal{A}(t,s^{n}x)\|
=log𝒜(t,x)1+log𝒜(sn,x)+log𝒜(t,snx)absentnorm𝒜superscript𝑡𝑥1norm𝒜superscript𝑠𝑛𝑥norm𝒜𝑡superscript𝑠𝑛𝑥\displaystyle=\log\|\mathcal{A}(t,x)^{-1}\|+\log\|\mathcal{A}(s^{n},x)\|+\log\|\mathcal{A}(t,s^{n}x)\|
logcm(x)+logcm(sn(x))+log𝒜(sn,x).absentsubscript𝑐𝑚𝑥subscript𝑐𝑚superscript𝑠𝑛𝑥norm𝒜superscript𝑠𝑛𝑥\displaystyle\leq\log c_{m}(x)+\log c_{m}(s^{n}(x))+\log\|\mathcal{A}(s^{n},x)\|.

Using that μ𝜇\mu is s𝑠s-invariant, we have for every n𝑛n that

log\displaystyle\int\log 𝒜(sn,x)d(Fmμ)(x)=1|Fm|Fmlog𝒜(sn,x)dtμ(x)𝑑tnorm𝒜superscript𝑠𝑛𝑥𝑑subscript𝐹𝑚𝜇𝑥1subscript𝐹𝑚subscriptsubscript𝐹𝑚norm𝒜superscript𝑠𝑛𝑥𝑑𝑡𝜇𝑥differential-d𝑡\displaystyle\|\mathcal{A}(s^{n},x)\|\ d(F_{m}\ast\mu)(x)=\dfrac{1}{|F_{m}|}\int_{F_{m}}\int\log\|\mathcal{A}(s^{n},x)\|\ dt*\mu(x)\ dt
=1|Fm|Fmlog𝒜(sn,tx)dμ(x)𝑑tabsent1subscript𝐹𝑚subscriptsubscript𝐹𝑚norm𝒜superscript𝑠𝑛𝑡𝑥𝑑𝜇𝑥differential-d𝑡\displaystyle=\dfrac{1}{|F_{m}|}\int_{F_{m}}\int\log\|\mathcal{A}(s^{n},tx)\|\ d\mu(x)\ dt
1|Fm|Fm(logcm(x)+logcm(sn(x))+log𝒜(sn,x)dμ(x))𝑑tabsent1subscript𝐹𝑚subscriptsubscript𝐹𝑚subscript𝑐𝑚𝑥subscript𝑐𝑚superscript𝑠𝑛𝑥norm𝒜superscript𝑠𝑛𝑥𝑑𝜇𝑥differential-d𝑡\displaystyle\leq\dfrac{1}{|F_{m}|}\int_{F_{m}}\bigg{(}\int\log c_{m}(x)+\log c_{m}(s^{n}(x))+\log\|\mathcal{A}(s^{n},x)\|\ d\mu(x)\bigg{)}\ dt
2logcm(x)𝑑μ(x)+log𝒜(sn,x)dμ(x)absent2subscript𝑐𝑚𝑥differential-d𝜇𝑥norm𝒜superscript𝑠𝑛𝑥𝑑𝜇𝑥\displaystyle\leq 2\int\log c_{m}(x)\ d\mu(x)+\int\log\|\mathcal{A}(s^{n},x)\|\ d\mu{(x)}

Dividing by n𝑛n yields λtop,s,Fmμ,𝒜λtop,s,μ,𝒜subscript𝜆top𝑠subscript𝐹𝑚𝜇𝒜subscript𝜆top𝑠𝜇𝒜\lambda_{\mathrm{top},s,F_{m}\ast\mu,\mathcal{A}}\leq\lambda_{\mathrm{top},s,\mu,\mathcal{A}}. The reverse inequality is similar.

The inequality then follows from the upper-semicontinuity in Lemma 3.8. ∎

Consider now any Y𝔤𝑌𝔤Y\in\mathfrak{g} with Y=1norm𝑌1\|Y\|=1, a point xX𝑥𝑋x\in X, and t>0𝑡0t>0. The empirical measure η(Y,t,x)𝜂𝑌𝑡𝑥\eta(Y,t,x) along the orbit exp(sY)x𝑠𝑌𝑥\exp(sY)x until time t𝑡t is the measure defined as follows: given a bounded continuous ϕ:X:italic-ϕ𝑋\phi\colon X\to\mathbb{R}, the integral of ϕitalic-ϕ\phi with respect to the empirical measure η(Y,t,x)𝜂𝑌𝑡𝑥\eta(Y,t,x) is

ϕ𝑑η(Y,t,x):=1t0tϕ(exp(sY)x)𝑑s.assignitalic-ϕdifferential-d𝜂𝑌𝑡𝑥1𝑡superscriptsubscript0𝑡italic-ϕ𝑠𝑌𝑥differential-d𝑠\int\phi\ d\eta(Y,t,x):=\frac{1}{t}\int_{0}^{t}\phi\big{(}\exp(sY)\cdot x\big{)}\ ds.

Similarly, given a probability measure μ𝜇\mu on X𝑋X, the empirical distribution η(Y,t,μ)𝜂𝑌𝑡𝜇\eta(Y,t,\mu) of μ𝜇\mu along the orbit of exp(sY)𝑠𝑌\exp(sY) until time t𝑡t is defined as

ϕ𝑑η(Y,t,μ):=1tX0tϕ(exp(sY)x)𝑑s𝑑μ(x).assignitalic-ϕdifferential-d𝜂𝑌𝑡𝜇1𝑡subscript𝑋superscriptsubscript0𝑡italic-ϕ𝑠𝑌𝑥differential-d𝑠differential-d𝜇𝑥\int\phi\ d\eta(Y,t,\mu):=\frac{1}{t}\int_{X}\int_{0}^{t}\phi\big{(}\exp(sY)\cdot x\big{)}\ ds\ d\mu(x).

Consider now sequences Yn𝔤subscript𝑌𝑛𝔤Y_{n}\in\mathfrak{g} with Yn=1normsubscript𝑌𝑛1\|Y_{n}\|=1 and tn>0subscript𝑡𝑛0t_{n}>0. For part (c) of the following lemma, we add an additional assumption that the action of G𝐺G on (X,d)𝑋𝑑(X,d) has uniform displacement: for any compact KG𝐾𝐺K\subset G there is Csuperscript𝐶C^{\prime} such that for all xX𝑥𝑋x\in X and gK𝑔𝐾g\in K,

d(gx,x)C.𝑑𝑔𝑥𝑥superscript𝐶d(g\cdot x,x)\leq C^{\prime}.
Lemma 3.10.

Suppose the action of G𝐺G on (X,d)𝑋𝑑(X,d) has uniform displacement and let 𝒜:G×:𝒜𝐺\mathcal{A}\colon G\times\mathcal{E}\to\mathcal{E} be a tempered continuous cocycle.

Let Yn𝔤subscript𝑌𝑛𝔤Y_{n}\in\mathfrak{g} and tn0subscript𝑡𝑛0t_{n}\geq 0 be sequences with Yn=1normsubscript𝑌𝑛1\|Y_{n}\|=1 for all n𝑛n and tnsubscript𝑡𝑛t_{n}\to\infty. Let μnsubscript𝜇𝑛\mu_{n} be a sequence of Borel probability measures on X𝑋X and define ηn:=η(Yn,tn,μn)assignsubscript𝜂𝑛𝜂subscript𝑌𝑛subscript𝑡𝑛subscript𝜇𝑛\eta_{n}:=\eta(Y_{n},t_{n},\mu_{n}) to be the empirical distribution of μnsubscript𝜇𝑛\mu_{n} along the orbit of exp(sYn)𝑠subscript𝑌𝑛\exp(sY_{n}) for 0stn0𝑠subscript𝑡𝑛0\leq s\leq t_{n}. Assume that

  1. (1)

    the family of empirical distributions {ηn}subscript𝜂𝑛\{\eta_{n}\} defined above has uniformly exponentially small mass in the cusps; and

  2. (2)

    log𝒜(exp(tnYn),x)dμn(x)εtnnorm𝒜subscript𝑡𝑛subscript𝑌𝑛𝑥𝑑subscript𝜇𝑛𝑥𝜀subscript𝑡𝑛\int\log\|\mathcal{A}(\exp(t_{n}Y_{n}),x)\|\ d\mu_{n}(x)\geq\varepsilon t_{n}.

Then

  1. (a)

    the family {ηn}subscript𝜂𝑛\{\eta_{n}\} is pre-compact;

  2. (b)

    for any subsequential limit Y=limjYnj,subscript𝑌subscript𝑗subscript𝑌subscript𝑛𝑗Y_{\infty}=\lim_{j\to\infty}Y_{n_{j}}, any subsequential limit ηsubscript𝜂\eta_{\infty} of {ηnj}subscript𝜂subscript𝑛𝑗\{\eta_{n_{j}}\} is invariant under the 1-parameter subgroup {exp(tY):t}conditional-set𝑡subscript𝑌𝑡\{\exp(tY_{\infty}):t\in\mathbb{R}\};

  3. (c)

    λtop,exp(Y),η,𝒜ε>0subscript𝜆topsubscript𝑌subscript𝜂𝒜𝜀0\lambda_{\mathrm{top},\exp(Y_{\infty}),\eta_{\infty},\mathcal{A}}\geq\varepsilon>0.

Proof of Lemma 3.10 (a) and (b).

As in the proof of Lemma 3.8, from the assumption that {ηn}subscript𝜂𝑛\{\eta_{n}\} has uniformly exponentially small mass in the cusps we obtain uniform bounds

ηn({x:d(x,x0)})Ceηsubscript𝜂𝑛conditional-set𝑥𝑑𝑥subscript𝑥0𝐶superscript𝑒𝜂\eta_{n}(\{x:d(x,x_{0})\geq\ell\})\leq Ce^{-\eta\ell}

for all n𝑛n. Combined with the properness of d𝑑d, this establishes uniform tightness of the family of measures {ηn}subscript𝜂𝑛\{\eta_{n}\} and (a) follows.

For (b), let ϕ:X:italic-ϕ𝑋\phi\colon X\to\mathbb{R} be a compactly supported continuous function. Then for any s>0𝑠0s>0

Xϕexp(sY)ϕdηnsubscript𝑋italic-ϕ𝑠subscript𝑌italic-ϕ𝑑subscript𝜂𝑛\displaystyle\int_{X}\phi\circ\exp(sY_{\infty})-\phi\ d\eta_{n} =Xϕexp(sY)ϕexp(sYn)dηnabsentsubscript𝑋italic-ϕ𝑠subscript𝑌italic-ϕ𝑠subscript𝑌𝑛𝑑subscript𝜂𝑛\displaystyle=\int_{X}\phi\circ\exp(sY_{\infty})-\phi\circ\exp(sY_{n})\ d\eta_{n}
+Xϕexp(sYn)ϕdηnsubscript𝑋italic-ϕ𝑠subscript𝑌𝑛italic-ϕ𝑑subscript𝜂𝑛\displaystyle+\int_{X}\phi\circ\exp(sY_{n})-\phi\ d\eta_{n}

The first integral converges to zero as the functions ϕexp(wY)ϕexp(wYn)italic-ϕ𝑤subscript𝑌italic-ϕ𝑤subscript𝑌𝑛\phi\circ\exp(wY_{\infty})-\phi\circ\exp(wY_{n}) converges uniformly to zero in n𝑛n for fixed w𝑤w. The second integral clearly converges to zero since for tnssubscript𝑡𝑛𝑠t_{n}\geq s we have

Xϕsubscript𝑋italic-ϕ\displaystyle\int_{X}\phi exp(sYn)ϕdηn=1tn0tnXϕ(exp((s+t)Yn)x)ϕ(exp(tYn)x)dμn(x)dt\displaystyle\circ\exp(sY_{n})-\phi\ d\eta_{n}=\frac{1}{t_{n}}\int_{0}^{t_{n}}\int_{X}\phi\left(\exp\left((s+t)Y_{n}\right)x\right)-\phi\left(\exp(tY_{n})x\right)\ d\mu_{n}(x)\ dt
=1tn[0sXϕ(exp(tYn)x)μn(x)𝑑t+tntn+sXϕ(exp(tYn)x)𝑑μn(x)𝑑t]absent1subscript𝑡𝑛delimited-[]superscriptsubscript0𝑠subscript𝑋italic-ϕ𝑡subscript𝑌𝑛𝑥subscript𝜇𝑛𝑥differential-d𝑡superscriptsubscriptsubscript𝑡𝑛subscript𝑡𝑛𝑠subscript𝑋italic-ϕ𝑡subscript𝑌𝑛𝑥differential-dsubscript𝜇𝑛𝑥differential-d𝑡\displaystyle=\frac{1}{t_{n}}\left[-\int_{0}^{s}\int_{X}\phi\left(\exp\left(tY_{n}\right)x\right)\mu_{n}(x)\ dt+\int_{t_{n}}^{t_{n}+s}\int_{X}\phi\left(\exp\left(tY_{n}\right)x\right)\ d\mu_{n}(x)\ dt\right]

which converges to 0 as tnsubscript𝑡𝑛t_{n}\to\infty as ϕitalic-ϕ\phi is bounded. ∎

The proof of Lemma 3.10(c) is quite involved. It is the analogue in the non-compact setting of [BFH, Lemma 3.6]; we recommend the reader read the proof of of [BFH, Lemma 3.6] first. Two technical complications arise in the proof of Lemma 3.10(c). First, we must control for “escape of Lyapunov exponent” as our cocycle is unbounded. Second, in [BFH] it was sufficient to consider the average of Dirac masses δxnsubscript𝛿subscript𝑥𝑛\delta_{x_{n}} along a single orbit exp(sYn)xn𝑠subscript𝑌𝑛subscript𝑥𝑛\exp(sY_{n})x_{n}; here we average measures μnsubscript𝜇𝑛\mu_{n} along an orbit of exp(sYn).𝑠subscript𝑌𝑛\exp(sY_{n}).

To prove Lemma 3.10(c) we first introduce a number of standard auxiliary objects. Let X𝑋\mathbb{P}\mathcal{E}\to X denote the projectivization of the tangent bundle X𝑋\mathcal{E}\to X. We represent a point in \mathbb{P}\mathcal{E} as (x,[v])𝑥delimited-[]𝑣(x,[v]) where [v]delimited-[]𝑣[v] is an equivalence class of non-zero vectors in the fiber (x)𝑥\mathcal{E}(x). For each n𝑛n, let σn:X{0}:subscript𝜎𝑛𝑋0\sigma_{n}\colon X\to\mathcal{E}\smallsetminus\{0\} be a nowhere vanishing Borel section such that

𝒜(exp(tnYn),x)(σn(x))(σn(x))1=𝒜(exp(tnYn),x)norm𝒜subscript𝑡𝑛subscript𝑌𝑛𝑥subscript𝜎𝑛𝑥superscriptnormsubscript𝜎𝑛𝑥1norm𝒜subscript𝑡𝑛subscript𝑌𝑛𝑥\|\mathcal{A}(\exp(t_{n}Y_{n}),x)(\sigma_{n}(x))\|\|(\sigma_{n}(x))\|^{-1}=\|\mathcal{A}(\exp(t_{n}Y_{n}),x)\|

for every xX𝑥𝑋x\in X. The G𝐺G-action on \mathcal{E} by vector-bundle automorphisms induces a natural G𝐺G-action on \mathbb{P}\mathcal{E} which restricts to projective transformations between each fiber and its image. For each n𝑛n, let η~nsubscript~𝜂𝑛\tilde{\eta}_{n} be the probability measure on \mathbb{P}\mathcal{E} given as follows: given a bounded continuous ϕ::italic-ϕ\phi\colon\mathbb{P}\mathcal{E}\to\mathbb{R} define

ϕ𝑑η~n:=1tn0tnXϕ(exp(tYn)(x,[σn(x)]))𝑑μn(x)𝑑t.assignsubscriptitalic-ϕdifferential-dsubscript~𝜂𝑛1subscript𝑡𝑛superscriptsubscript0subscript𝑡𝑛subscript𝑋italic-ϕ𝑡subscript𝑌𝑛𝑥delimited-[]subscript𝜎𝑛𝑥differential-dsubscript𝜇𝑛𝑥differential-d𝑡\int_{\mathbb{P}\mathcal{E}}\phi\ d\tilde{\eta}_{n}:=\frac{1}{t_{n}}\int_{0}^{t_{n}}\int_{X}\phi\big{(}\exp(tY_{n})\cdot(x,[\sigma_{n}(x)])\big{)}\ d\mu_{n}(x)\ dt.

We have that η~nsubscript~𝜂𝑛\tilde{\eta}_{n} projects to ηnsubscript𝜂𝑛\eta_{n} under the natural projection X𝑋\mathbb{P}\mathcal{E}\to X; moreover, if ηjksubscript𝜂subscript𝑗𝑘\eta_{j_{k}} is a subsequence converging to η,subscript𝜂\eta_{\infty}, then any weak-* subsequential limit η~subscript~𝜂\tilde{\eta}_{\infty} of {η~njk}subscript~𝜂subscript𝑛subscript𝑗𝑘\{\tilde{\eta}_{n_{j_{k}}}\} projects to ηsubscript𝜂\eta_{\infty}.

Define Φ:𝔤×:Φ𝔤\Phi\colon\mathfrak{g}\times\mathbb{P}\mathcal{E}\to\mathbb{R} by

Φ(Y,(x,[v])):=log(𝒜(exp(Y),x)vv1).assignΦ𝑌𝑥delimited-[]𝑣norm𝒜𝑌𝑥𝑣superscriptnorm𝑣1\Phi\big{(}Y,(x,[v])\big{)}:=\log\left(\left\|\mathcal{A}\big{(}\exp(Y),x\big{)}v\right\|\|v\|^{-1}\right).

Note for each fixed Y𝔤𝑌𝔤Y\in\mathfrak{g} that ΦΦ\Phi satisfies a cocycle property:

Φ((s+t)Y,(x,[v]))=Φ(tY,(x,[v]))+Φ(sY,exp(tY)(x,[v]))Φ𝑠𝑡𝑌𝑥delimited-[]𝑣Φ𝑡𝑌𝑥delimited-[]𝑣Φ𝑠𝑌𝑡𝑌𝑥delimited-[]𝑣\Phi\big{(}(s+t)Y,(x,[v])\big{)}=\Phi\big{(}tY,(x,[v])\big{)}+\Phi\big{(}sY,\exp(tY)\cdot(x,[v])\big{)} (13)

By hypothesis, there are C>1𝐶1C>1, k1𝑘1k\geq 1, and η>0𝜂0\eta>0 such that

eηd(x,x0)𝑑ηnCsuperscript𝑒𝜂𝑑𝑥subscript𝑥0differential-dsubscript𝜂𝑛𝐶\int e^{\eta d(x,x_{0})}\ d\eta_{n}\leq C

for all n𝑛n and

1Cekd(x,x0)𝒜(exp(Y),x)vv1Cekd(x,x0)1𝐶superscript𝑒𝑘𝑑𝑥subscript𝑥0norm𝒜𝑌𝑥𝑣superscriptnorm𝑣1𝐶superscript𝑒𝑘𝑑𝑥subscript𝑥0\frac{1}{C}e^{-kd(x,x_{0})}\leq\left\|\mathcal{A}(\exp(Y),x)v\right\|\|v\|^{-1}\leq Ce^{kd(x,x_{0})}

for all (x,[v])𝑥delimited-[]𝑣(x,[v])\in\mathbb{P}\mathcal{E} and Y𝔤𝑌𝔤Y\in\mathfrak{g} with Y1.norm𝑌1\|Y\|\leq 1.

For each n𝑛n, let

Mn(x)=sup0ttn{d((exp(tYn)x),x0)}.subscript𝑀𝑛𝑥subscriptsupremum0𝑡subscript𝑡𝑛𝑑𝑡subscript𝑌𝑛𝑥subscript𝑥0M_{n}(x)=\sup_{0\leq t\leq t_{n}}\left\{d\left(\big{(}\exp(tY_{n})x\big{)},x_{0}\right)\right\}.

As we assume the G𝐺G-action on (X,d)𝑋𝑑(X,d) has uniform displacement, take

C1=supY1,xX{d(exp(Y)x,x)}.subscript𝐶1subscriptsupremumformulae-sequencenorm𝑌1𝑥𝑋𝑑𝑌𝑥𝑥C_{1}=\sup_{\|Y\|\leq 1,x\in X}\{d(\exp(Y)\cdot x,x)\}.

We have

1tn0tnXeηd((exp(tYn)x,x0)𝑑μn(x)𝑑t=eηd(x,x0)𝑑ηnC.\frac{1}{t_{n}}\int_{0}^{t_{n}}\int_{X}e^{\eta d\left((\exp(tY_{n})x,x_{0}\right)}\ d\mu_{n}(x)\ dt=\int e^{\eta d(x,x_{0})}\ d\eta_{n}\leq C.

If tn1subscript𝑡𝑛1t_{n}\geq 1 then for every x𝑥x there is an interval Ix[0,tn]subscript𝐼𝑥0subscript𝑡𝑛I_{x}\subset[0,t_{n}] of length 111 on which

d((exp(tYn)x,x0)(Mn(x)C1)d\left((\exp(tY_{n})x,x_{0}\right)\geq(M_{n}(x)-C_{1})

for all tIx.𝑡subscript𝐼𝑥t\in I_{x}. It follows that

Xeη(Mn(x)C1)𝑑μn(x)XIxeηd((exp(tYn)x,x0)𝑑t𝑑μn(x)Ctn.\int_{X}e^{\eta(M_{n}(x)-C_{1})}\ d\mu_{n}(x)\leq\int_{X}\int_{I_{x}}e^{\eta d\left((\exp(tY_{n})x,x_{0}\right)}\ dt\ d\mu_{n}(x)\leq Ct_{n}.

By Jensen’s inequality we have

Xη(Mn(x)C1)𝑑μn(x)logXeη(Mn(x)C1)𝑑μn(x)subscript𝑋𝜂subscript𝑀𝑛𝑥subscript𝐶1differential-dsubscript𝜇𝑛𝑥subscript𝑋superscript𝑒𝜂subscript𝑀𝑛𝑥subscript𝐶1differential-dsubscript𝜇𝑛𝑥\int_{X}\eta(M_{n}(x)-C_{1})\ d\mu_{n}(x)\leq\log\int_{X}e^{\eta(M_{n}(x)-C_{1})}d\mu_{n}(x)

whence

Mn(x)dμn(x)η1(logC+logtn)+C1=:η1logtn+C2.\int M_{n}(x)\ d\mu_{n}(x)\leq\eta^{-1}(\log C+\log t_{n})+C_{1}=:\eta^{-1}\log t_{n}+C_{2}.

Since Yn=1,normsubscript𝑌𝑛1\|Y_{n}\|=1, we have

sup0ttn,0s1Xsubscriptsupremumformulae-sequence0𝑡subscript𝑡𝑛0𝑠1subscript𝑋\displaystyle\sup_{0\leq t\leq t_{n},0\leq s\leq 1}\int_{X} |Φ(sYn,exp(tYn)(x,[σn(x)]))|dμn(x)Φ𝑠subscript𝑌𝑛𝑡subscript𝑌𝑛𝑥delimited-[]subscript𝜎𝑛𝑥𝑑subscript𝜇𝑛𝑥\displaystyle\left|\Phi(sY_{n},\exp(tY_{n})\cdot(x,[\sigma_{n}(x)]))\right|\ d\mu_{n}(x) (14)
Xsup0ttn,0s1|Φ(sYn,exp(tYn)(x,[σn(x)]))|dμn(x)absentsubscript𝑋subscriptsupremumformulae-sequence0𝑡subscript𝑡𝑛0𝑠1Φ𝑠subscript𝑌𝑛𝑡subscript𝑌𝑛𝑥delimited-[]subscript𝜎𝑛𝑥𝑑subscript𝜇𝑛𝑥\displaystyle\leq\int_{X}\sup_{0\leq t\leq t_{n},0\leq s\leq 1}\left|\Phi(sY_{n},\exp(tY_{n})\cdot(x,[\sigma_{n}(x)]))\right|\ d\mu_{n}(x)
|logC|+kMn(x)dμn(x)absent𝐶𝑘subscript𝑀𝑛𝑥𝑑subscript𝜇𝑛𝑥\displaystyle\leq\int|\log C|+kM_{n}(x)\ d\mu_{n}(x)
|logC|+k(η1logtn+C2)absent𝐶𝑘superscript𝜂1subscript𝑡𝑛subscript𝐶2\displaystyle\leq|\log C|+k(\eta^{-1}\log t_{n}+C_{2})
=:kη1logtn+C3.\displaystyle=:k\eta^{-1}\log t_{n}+C_{3}.

In particular, we have

1tn1subscript𝑡𝑛\displaystyle\frac{1}{t_{n}} Xlog𝒜(exp(tnYn),x)dμn(x)subscript𝑋norm𝒜subscript𝑡𝑛subscript𝑌𝑛𝑥𝑑subscript𝜇𝑛𝑥\displaystyle\int_{X}\log\|\mathcal{A}(\exp(t_{n}Y_{n}),x)\|\ d\mu_{n}(x)
=1tnXΦ(tnYn,(x,[σn(x)]))𝑑μn(x)absent1subscript𝑡𝑛subscript𝑋Φsubscript𝑡𝑛subscript𝑌𝑛𝑥delimited-[]subscript𝜎𝑛𝑥differential-dsubscript𝜇𝑛𝑥\displaystyle=\frac{1}{t_{n}}\int_{X}\Phi(t_{n}Y_{n},(x,[\sigma_{n}(x)]))\ d\mu_{n}(x)
=1tnXΦ(tnYn,(x,[σn(x)]))𝑑μn(x)absent1subscript𝑡𝑛subscript𝑋Φsubscript𝑡𝑛subscript𝑌𝑛𝑥delimited-[]subscript𝜎𝑛𝑥differential-dsubscript𝜇𝑛𝑥\displaystyle=\frac{1}{t_{n}}\int_{X}\Phi(\lfloor t_{n}\rfloor Y_{n},(x,[\sigma_{n}(x)]))\ d\mu_{n}(x)
+1tnXΦ((ttn)Yn,exp(tnYn)(x,[σn(x)]))𝑑μn(x).1subscript𝑡𝑛subscript𝑋Φ𝑡subscript𝑡𝑛subscript𝑌𝑛subscript𝑡𝑛subscript𝑌𝑛𝑥delimited-[]subscript𝜎𝑛𝑥differential-dsubscript𝜇𝑛𝑥\displaystyle\quad+\frac{1}{t_{n}}\int_{X}\Phi((t-\lfloor t_{n}\rfloor)Y_{n},\exp(\lfloor t_{n}\rfloor Y_{n})\cdot(x,[\sigma_{n}(x)]))\ d\mu_{n}(x).

Since

|1tnXΦ((ttn)Yn,exp(tnYn)(x,[σn(x)]))𝑑μn(x)|1tn(kη1logtn+C3)1subscript𝑡𝑛subscript𝑋Φ𝑡subscript𝑡𝑛subscript𝑌𝑛subscript𝑡𝑛subscript𝑌𝑛𝑥delimited-[]subscript𝜎𝑛𝑥differential-dsubscript𝜇𝑛𝑥1subscript𝑡𝑛𝑘superscript𝜂1subscript𝑡𝑛subscript𝐶3\left|\frac{1}{t_{n}}\int_{X}\Phi((t-\lfloor t_{n}\rfloor)Y_{n},\exp(\lfloor t_{n}\rfloor Y_{n})\cdot(x,[\sigma_{n}(x)]))\ d\mu_{n}(x)\right|\leq\frac{1}{t_{n}}(k\eta^{-1}\log t_{n}+C_{3})

goes to 0 as tnsubscript𝑡𝑛t_{n}\to\infty it follows that

lim infnX1tnsubscriptlimit-infimum𝑛subscript𝑋1subscript𝑡𝑛\displaystyle\liminf_{n\to\infty}\int_{X}\frac{1}{t_{n}} Φ(tnYn,(x,[σn(x)]))dμn(x)Φsubscript𝑡𝑛subscript𝑌𝑛𝑥delimited-[]subscript𝜎𝑛𝑥𝑑subscript𝜇𝑛𝑥\displaystyle\Phi(\lfloor t_{n}\rfloor Y_{n},(x,[\sigma_{n}(x)]))\ d\mu_{n}(x) (15)
=lim infn1tnXlog𝒜(exp(tnYn),x)dμn(x)absentsubscriptlimit-infimum𝑛1subscript𝑡𝑛subscript𝑋norm𝒜subscript𝑡𝑛subscript𝑌𝑛𝑥𝑑subscript𝜇𝑛𝑥\displaystyle=\liminf_{n\to\infty}\frac{1}{t_{n}}\int_{X}\log\|\mathcal{A}(\exp(t_{n}Y_{n}),x)\|\ d\mu_{n}(x)
ε>0.absent𝜀0\displaystyle\geq\varepsilon>0.

With the above objects and estimates we complete the proof of Lemma 3.10.

Proof of Lemma 3.10 (c).

Consider first the expression Φ(Yn,)𝑑η~n.Φsubscript𝑌𝑛differential-dsubscript~𝜂𝑛\int\Phi(Y_{n},\cdot)\ d\tilde{\eta}_{n}. We have

ΦΦ\displaystyle\int\Phi (Yn,)dη~nsubscript𝑌𝑛𝑑subscript~𝜂𝑛\displaystyle(Y_{n},\cdot)\ d\tilde{\eta}_{n}
=1tn0tnXΦ(Yn,exp(tYn)(x,[σn(x)]))𝑑μn(x)𝑑tabsent1subscript𝑡𝑛superscriptsubscript0subscript𝑡𝑛subscript𝑋Φsubscript𝑌𝑛𝑡subscript𝑌𝑛𝑥delimited-[]subscript𝜎𝑛𝑥differential-dsubscript𝜇𝑛𝑥differential-d𝑡\displaystyle=\frac{1}{t_{n}}\int_{0}^{t_{n}}\int_{X}\Phi\big{(}Y_{n},\exp(tY_{n})\cdot(x,[\sigma_{n}(x)])\big{)}\ d\mu_{n}(x)\ dt
=1tn0tnXΦ(Yn,exp(tYn)(x,[σn(x)]))𝑑μn(x)𝑑tabsent1subscript𝑡𝑛superscriptsubscript0subscript𝑡𝑛subscript𝑋Φsubscript𝑌𝑛𝑡subscript𝑌𝑛𝑥delimited-[]subscript𝜎𝑛𝑥differential-dsubscript𝜇𝑛𝑥differential-d𝑡\displaystyle=\frac{1}{t_{n}}\int_{0}^{\lfloor t_{n}\rfloor}\int_{X}\Phi\big{(}Y_{n},\exp(tY_{n})\cdot(x,[\sigma_{n}(x)])\big{)}\ d\mu_{n}(x)\ dt
+1tntntnXΦ(Yn,exp(tYn)(x,[σn(x)]))𝑑μn(x)𝑑t1subscript𝑡𝑛superscriptsubscriptsubscript𝑡𝑛subscript𝑡𝑛subscript𝑋Φsubscript𝑌𝑛𝑡subscript𝑌𝑛𝑥delimited-[]subscript𝜎𝑛𝑥differential-dsubscript𝜇𝑛𝑥differential-d𝑡\displaystyle\quad+\frac{1}{t_{n}}\int_{\lfloor t_{n}\rfloor}^{t_{n}}\int_{X}\Phi\big{(}Y_{n},\exp(tY_{n})\cdot(x,[\sigma_{n}(x)])\big{)}\ d\mu_{n}(x)\ dt

Note that the contribution of the second integral is bounded by

|1tntntnXΦ(Yn,exp(tYn)(x,[σn(x)]))𝑑μn(x)𝑑t|1tn(kη1logtn+C3)1subscript𝑡𝑛superscriptsubscriptsubscript𝑡𝑛subscript𝑡𝑛subscript𝑋Φsubscript𝑌𝑛𝑡subscript𝑌𝑛𝑥delimited-[]subscript𝜎𝑛𝑥differential-dsubscript𝜇𝑛𝑥differential-d𝑡1subscript𝑡𝑛𝑘superscript𝜂1subscript𝑡𝑛subscript𝐶3\left|\frac{1}{t_{n}}\int_{\lfloor t_{n}\rfloor}^{t_{n}}\int_{X}\Phi\big{(}Y_{n},\exp(tY_{n})\cdot(x,[\sigma_{n}(x)])\big{)}\ d\mu_{n}(x)\ dt\right|\leq\frac{1}{t_{n}}(k\eta^{-1}\log t_{n}+C_{3})

which goes to zero as tnsubscript𝑡𝑛t_{n}\to\infty.

Repeatedly applying the cocycle property (13) of Φ(Yn,)Φsubscript𝑌𝑛\Phi(Y_{n},\cdot) we have for tn1subscript𝑡𝑛1t_{n}\geq 1 that

1tnX0tnΦ1subscript𝑡𝑛subscript𝑋superscriptsubscript0subscript𝑡𝑛Φ\displaystyle\frac{1}{t_{n}}\int_{X}\int_{0}^{\lfloor t_{n}\rfloor}\Phi (Yn,exp(tYn)(x,[σn(x)]))dtdμn(x)subscript𝑌𝑛𝑡subscript𝑌𝑛𝑥delimited-[]subscript𝜎𝑛𝑥𝑑𝑡𝑑subscript𝜇𝑛𝑥\displaystyle\big{(}Y_{n},\exp(tY_{n})\cdot(x,[\sigma_{n}(x)])\big{)}\ dt\ d\mu_{n}(x)
=1tnX01Φ(tnYn,exp(tYn)(x,[σn(x)]))𝑑t𝑑μn(x)absent1subscript𝑡𝑛subscript𝑋superscriptsubscript01Φsubscript𝑡𝑛subscript𝑌𝑛𝑡subscript𝑌𝑛𝑥delimited-[]subscript𝜎𝑛𝑥differential-d𝑡differential-dsubscript𝜇𝑛𝑥\displaystyle=\frac{1}{t_{n}}\int_{X}\int_{0}^{1}\Phi\big{(}{\lfloor t_{n}\rfloor}Y_{n},\exp(tY_{n})\cdot(x,[\sigma_{n}(x)])\big{)}\ dt\ d\mu_{n}(x)
=1tnX01(Φ(tnYn,(x,[σn(x)]))Φ(tYn,(x,[σn(x)]))\displaystyle=\frac{1}{t_{n}}\int_{X}\int_{0}^{1}\Big{(}\Phi\big{(}{\lfloor t_{n}\rfloor}Y_{n},(x,[\sigma_{n}(x)])\big{)}-\Phi\big{(}tY_{n},(x,[\sigma_{n}(x)])\big{)}
+Φ(tYn,exp(tnYn)(x,[σn(x)])))dtdμn(x)\displaystyle\quad\quad\quad+\Phi\big{(}tY_{n},\exp(\lfloor t_{n}\rfloor Y_{n})\cdot(x,[\sigma_{n}(x)])\big{)}\Big{)}\ dt\ d\mu_{n}(x)
=1tnXΦ(tnYn,(x,[σn(x)])dμn(x)+1tnX01(Φ(tYn,(x,[σn(x)]))\displaystyle=\frac{1}{t_{n}}\int_{X}\Phi\big{(}{\lfloor t_{n}\rfloor}Y_{n},(x,[\sigma_{n}(x)])\ d\mu_{n}(x)+\frac{1}{t_{n}}\int_{X}\int_{0}^{1}\Big{(}-\Phi\big{(}tY_{n},(x,[\sigma_{n}(x)])\big{)}
+Φ(tYn,exp(tnYn)(x,[σn(x)])))dtdμn(x)\displaystyle\quad\quad\quad+\Phi\big{(}tY_{n},\exp(\lfloor t_{n}\rfloor Y_{n})\cdot(x,[\sigma_{n}(x)])\big{)}\Big{)}\ dt\ d\mu_{n}(x)

From (14), the contribution of the second and third integrals is bounded by

|1tnX01(Φ(tYn,(x,[σn(x)]))+Φ(tYn,exp(tnYn)(x,[σn(x)])))𝑑t𝑑μn(x)|1subscript𝑡𝑛subscript𝑋superscriptsubscript01Φ𝑡subscript𝑌𝑛𝑥delimited-[]subscript𝜎𝑛𝑥Φ𝑡subscript𝑌𝑛subscript𝑡𝑛subscript𝑌𝑛𝑥delimited-[]subscript𝜎𝑛𝑥differential-d𝑡differential-dsubscript𝜇𝑛𝑥\displaystyle\left|\frac{1}{t_{n}}\int_{X}\int_{0}^{1}\Big{(}-\Phi\big{(}tY_{n},(x,[\sigma_{n}(x)])\big{)}+\Phi\big{(}tY_{n},\exp(\lfloor t_{n}\rfloor Y_{n})\cdot(x,[\sigma_{n}(x)])\big{)}\Big{)}\ dt\ d\mu_{n}(x)\right|
1tn012(kη1logtn+C3)𝑑tabsent1subscript𝑡𝑛superscriptsubscript012𝑘superscript𝜂1subscript𝑡𝑛subscript𝐶3differential-d𝑡\displaystyle\quad\quad\quad\quad\leq\frac{1}{t_{n}}\int_{0}^{1}2(k\eta^{-1}\log t_{n}+C_{3})\ dt
=1tn2(kη1logtn+C3)absent1subscript𝑡𝑛2𝑘superscript𝜂1subscript𝑡𝑛subscript𝐶3\displaystyle\quad\quad\quad\quad=\frac{1}{t_{n}}2(k\eta^{-1}\log t_{n}+C_{3})

which tend to zero as tnsubscript𝑡𝑛t_{n}\to\infty. We then conclude from (15) that

lim infnΦ(Yn,)dη~n=lim infn1tnXΦ(tnYn,(x,[σn(x)])dμn(x)ε>0.\liminf_{n\to\infty}\int\Phi(Y_{n},\cdot)\ d\tilde{\eta}_{n}=\liminf_{n\to\infty}\frac{1}{t_{n}}\int_{X}\Phi\big{(}{\lfloor t_{n}\rfloor}Y_{n},(x,[\sigma_{n}(x)])\ d\mu_{n}(x)\geq\varepsilon>0. (16)

To complete the proof of (c), for M>0𝑀0M>0 take ψM:X[0,1]:subscript𝜓𝑀𝑋01\psi_{M}\colon X\to[0,1] continuous with

ψM(x)=1 if d(x,x0)M and ψM(x)=0 if d(x,x0)M+1.ψM(x)=1 if d(x,x0)M and ψM(x)=0 if d(x,x0)M+1\text{$\psi_{M}(x)=1$ if $d(x,x_{0})\leq M$ and $\psi_{M}(x)=0$ if $d(x,x_{0})\geq M+1$}.

Let ΨM:[0,1]:subscriptΨ𝑀01\Psi_{M}\colon\mathbb{P}\mathcal{E}\to[0,1] be

ΨM(x,[v])=ψM(x).subscriptΨ𝑀𝑥delimited-[]𝑣subscript𝜓𝑀𝑥\Psi_{M}(x,[v])=\psi_{M}(x).

and define ΦM:𝔤×:subscriptΦ𝑀𝔤\Phi_{M}\colon\mathfrak{g}\times\mathbb{P}\mathcal{E}\to\mathbb{R} to be

ΦM(Y,(x,[v])):=ΨM(x,[v])Φ(Y,(x,[v])).assignsubscriptΦ𝑀𝑌𝑥delimited-[]𝑣subscriptΨ𝑀𝑥delimited-[]𝑣Φ𝑌𝑥delimited-[]𝑣\Phi_{M}\big{(}Y,(x,[v])\big{)}:=\Psi_{M}(x,[v])\Phi\big{(}Y,(x,[v])\big{)}.

As the family

𝒩={ηn}{η}𝒩subscript𝜂𝑛subscript𝜂\mathcal{N}=\{\eta_{n}\}\cup\{\eta_{\infty}\}

has uniformly exponentially small mass in the cusps we have

eηd(x,x0)𝑑η^<Csuperscript𝑒𝜂𝑑𝑥subscript𝑥0differential-d^𝜂𝐶\int e^{\eta d(x,x_{0})}d\hat{\eta}<C

and hence η^{x:d(x,x0)}Ceη^𝜂conditional-set𝑥𝑑𝑥subscript𝑥0𝐶superscript𝑒𝜂\hat{\eta}\{x:d(x,x_{0})\geq\ell\}\leq Ce^{-\eta\ell} for all η^𝒩^𝜂𝒩\hat{\eta}\in\mathcal{N}. It follows for all η~{η~n}{η~}~𝜂subscript~𝜂𝑛subscript~𝜂\tilde{\eta}\in\{\tilde{\eta}_{n}\}\cup\{\tilde{\eta}_{\infty}\} that—letting η^𝒩^𝜂𝒩\hat{\eta}\in\mathcal{N} denote the image of η~~𝜂\tilde{\eta} in X𝑋X—we have for any Y𝔤𝑌𝔤Y\in\mathfrak{g} with Y1norm𝑌1\|Y\|\leq 1 that

subscript\displaystyle\int_{\mathbb{P}\mathcal{E}} |Φ(Y,)ΦM(Y,)|dη~Φ𝑌subscriptΦ𝑀𝑌𝑑~𝜂\displaystyle|\Phi(Y,\cdot)-\Phi_{M}(Y,\cdot)|\ d\tilde{\eta}
={(x,[v]:d(x,x0)M}|Φ(Y,)ΦM(Y,)|𝑑η~\displaystyle=\int_{\{(x,[v]\in\mathbb{P}\mathcal{E}:d(x,x_{0})\geq M\}}|\Phi(Y,\cdot)-\Phi_{M}(Y,\cdot)|\ d{\tilde{\eta}}
{(x,[v]:d(x,x0)M}|Φ(Y,)|𝑑η~\displaystyle\leq\int_{\{(x,[v]\in\mathbb{P}\mathcal{E}:d(x,x_{0})\geq M\}}|\Phi(Y,\cdot)|\ d{\tilde{\eta}}
{xX:d(x,x0)M}log(C)+kd(x,x0)dη^absentsubscriptconditional-set𝑥𝑋𝑑𝑥subscript𝑥0𝑀𝐶𝑘𝑑𝑥subscript𝑥0𝑑^𝜂\displaystyle\leq\int_{\{x\in X:d(x,x_{0})\geq M\}}\log(C)+{kd(x,x_{0})}\ d\hat{\eta}
(logC+kM)CeηM+kMη^{x:d(x,x0)}𝑑absent𝐶𝑘𝑀𝐶superscript𝑒𝜂𝑀𝑘superscriptsubscript𝑀^𝜂conditional-set𝑥𝑑𝑥subscript𝑥0differential-d\displaystyle\leq{(\log C+kM)Ce^{-\eta M}+k\int_{M}^{\infty}\hat{\eta}\{x:{d(x,x_{0})}\geq\ell\}\ d\ell}
(logC+kM)CeηM+kCeη(M)η.absent𝐶𝑘𝑀𝐶superscript𝑒𝜂𝑀𝑘𝐶superscript𝑒𝜂𝑀𝜂\displaystyle\leq(\log C+kM)Ce^{-\eta M}+k\frac{Ce^{\eta(-M)}}{\eta}.

In particular, given any δ>0𝛿0\delta>0, by taking M>0𝑀0M>0 sufficiently large we may ensure that

|Φ(Y,)ΦM(Y,)|𝑑η~δsubscriptΦ𝑌subscriptΦ𝑀𝑌differential-d~𝜂𝛿\int_{\mathbb{P}\mathcal{E}}|\Phi(Y,\cdot)-\Phi_{M}(Y,\cdot)|\ d\tilde{\eta}\leq\delta

for any

η~{η~n}{η~}.~𝜂subscript~𝜂𝑛subscript~𝜂\tilde{\eta}\in\{\tilde{\eta}_{n}\}\cup\{\tilde{\eta}_{\infty}\}.

Since the restriction of ΦMsubscriptΦ𝑀\Phi_{M} to {Y𝔤:Y1}×conditional-set𝑌𝔤norm𝑌1\{Y\in\mathfrak{g}:\|Y\|\leq 1\}\times\mathbb{P}\mathcal{E} is compactly supported, it is uniformly continuous whence

ΦM(Yn,)𝑑η~nΦM(Y,)dη~0subscriptΦ𝑀subscript𝑌𝑛differential-dsubscript~𝜂𝑛subscriptΦ𝑀subscript𝑌𝑑subscript~𝜂0\int\Phi_{M}(Y_{n},\cdot)\ d\tilde{\eta}_{n}-\Phi_{M}(Y_{\infty},\cdot)\ d\tilde{\eta}_{\infty}\to 0

as n.𝑛n\to\infty. In particular given δ>0𝛿0\delta>0 we may take M𝑀M and n𝑛n sufficiently large so that

|\displaystyle\Big{|}\int_{\mathbb{P}\mathcal{E}} Φ(Yn,)dη~nΦ(Y,)dη~|\displaystyle\Phi(Y_{n},\cdot)\ d\tilde{\eta}_{n}-\int_{\mathbb{P}\mathcal{E}}\Phi(Y_{\infty},\cdot)\ d\tilde{\eta}_{\infty}\Big{|}
|Φ(Yn,)ΦM(Yn,)|𝑑η~nabsentsubscriptΦsubscript𝑌𝑛subscriptΦ𝑀subscript𝑌𝑛differential-dsubscript~𝜂𝑛\displaystyle\leq\int_{\mathbb{P}\mathcal{E}}\left|\Phi(Y_{n},\cdot)-\Phi_{M}(Y_{n},\cdot)\right|d\tilde{\eta}_{n}
+|ΦM(Yn,)ΦM(Y,)|𝑑η~nsubscriptsubscriptΦ𝑀subscript𝑌𝑛subscriptΦ𝑀subscript𝑌differential-dsubscript~𝜂𝑛\displaystyle\quad\quad+\int_{\mathbb{P}\mathcal{E}}\left|\Phi_{M}(Y_{n},\cdot)-\Phi_{M}(Y_{\infty},\cdot)\right|d\tilde{\eta}_{n}
+|Φ(Y,)ΦM(Y,)|𝑑η~subscriptΦsubscript𝑌subscriptΦ𝑀subscript𝑌differential-dsubscript~𝜂\displaystyle\quad\quad+\int_{\mathbb{P}\mathcal{E}}\left|\Phi(Y_{\infty},\cdot)-\Phi_{M}(Y_{\infty},\cdot)\right|d\tilde{\eta}_{\infty}
3δ.absent3𝛿\displaystyle\leq 3\delta.

Let g=exp(Y)subscript𝑔subscript𝑌g_{\infty}=\exp(Y_{\infty}). Note for each n𝑛n that

Xlog𝒜(gn,x)dη(x)log(𝒜(gn,x)vv1)𝑑η~(x,[v]).subscript𝑋norm𝒜superscriptsubscript𝑔𝑛𝑥𝑑subscript𝜂𝑥subscriptnorm𝒜superscriptsubscript𝑔𝑛𝑥𝑣superscriptnorm𝑣1differential-dsubscript~𝜂𝑥delimited-[]𝑣\int_{X}\log\|\mathcal{A}(g_{\infty}^{n},x)\|\ d\eta_{\infty}(x)\geq\int_{\mathbb{P}\mathcal{E}}\log(\left\|\mathcal{A}(g_{\infty}^{n},x)v\right\|\|v\|^{-1})\ d\tilde{\eta}_{\infty}(x,[v]).

It then follows for any δ>0𝛿0\delta>0

λtop,g,η,𝒜subscript𝜆topsubscript𝑔𝜂𝒜\displaystyle\lambda_{\mathrm{top},g_{\infty},\eta,\mathcal{A}} =limn1nXlog𝒜(gn,x)dη(x)absentsubscript𝑛1𝑛subscript𝑋norm𝒜superscriptsubscript𝑔𝑛𝑥𝑑subscript𝜂𝑥\displaystyle=\lim_{n\to\infty}\frac{1}{n}\int_{X}\log\|\mathcal{A}(g_{\infty}^{n},x)\|\ d\eta_{\infty}(x)
lim infn1nlog(𝒜(gn,x)vv1)𝑑η~(x,[v])absentsubscriptlimit-infimum𝑛1𝑛subscriptnorm𝒜superscriptsubscript𝑔𝑛𝑥𝑣superscriptnorm𝑣1differential-dsubscript~𝜂𝑥delimited-[]𝑣\displaystyle\geq\liminf_{n\to\infty}\frac{1}{n}\int_{\mathbb{P}\mathcal{E}}\log\left(\left\|\mathcal{A}(g_{\infty}^{n},x)v\right\|\|v\|^{-1}\right)\ d\tilde{\eta}_{\infty}(x,[v])
=lim infn1nΦ(nY,(x,[v]))𝑑η~(x,[v])absentsubscriptlimit-infimum𝑛1𝑛subscriptΦ𝑛subscript𝑌𝑥delimited-[]𝑣differential-dsubscript~𝜂𝑥delimited-[]𝑣\displaystyle=\liminf_{n\to\infty}\frac{1}{n}\int_{\mathbb{P}\mathcal{E}}\Phi(nY_{\infty},(x,[v]))\ d\tilde{\eta}_{\infty}(x,[v])
=Φ(Y,(x,[v]))𝑑η~(x,[v])absentsubscriptΦsubscript𝑌𝑥delimited-[]𝑣differential-dsubscript~𝜂𝑥delimited-[]𝑣\displaystyle=\int_{\mathbb{P}\mathcal{E}}\Phi(Y_{\infty},(x,[v]))\ d\tilde{\eta}_{\infty}(x,[v])
lim infnΦ(Yn,)𝑑η~n3δ.absentsubscriptlimit-infimum𝑛subscriptΦsubscript𝑌𝑛differential-dsubscript~𝜂𝑛3𝛿\displaystyle\geq\liminf_{n\to\infty}\int_{\mathbb{P}\mathcal{E}}\Phi(Y_{n},\cdot)\ d\tilde{\eta}_{n}-3\delta.

where the third equality follows from the invariance of η~subscript~𝜂\tilde{\eta}_{\infty} and the cocycle property of ΦΦ\Phi. Since

lim infnΦ(Yn,)𝑑η~nεsubscriptlimit-infimum𝑛subscriptΦsubscript𝑌𝑛differential-dsubscript~𝜂𝑛𝜀\liminf_{n\to\infty}\int_{\mathbb{P}\mathcal{E}}\Phi(Y_{n},\cdot)\ d\tilde{\eta}_{n}\geq\varepsilon

we conclude that

λtop,g,η,𝒜ε3δsubscript𝜆topsubscript𝑔𝜂𝒜𝜀3𝛿\lambda_{\mathrm{top},g_{\infty},\eta,\mathcal{A}}\geq\varepsilon-3\delta

for any δ>0𝛿0\delta>0 whence the result follows. ∎

3.7. Oseledec’s theorem for cocycles over actions by higher-rank abelian groups

Let AG𝐴𝐺A\subset G be a split Cartan subgroup. Then Adsimilar-to-or-equals𝐴superscript𝑑A\simeq\mathbb{R}^{d} where d𝑑d is the rank of G𝐺G. We have the following consequence of the higher-rank Oseledec’s multiplicative ergodic theorem (c.f. [BRH, Theorem 2.4]).

Fix any norm |||\cdot| on Adsimilar-to-or-equals𝐴superscript𝑑A\simeq\mathbb{R}^{d} and let η:X:𝜂𝑋\eta\colon X\to\mathbb{R} be

η(x):=sup|a|1log𝒜(a,x).assign𝜂𝑥subscriptsupremum𝑎1norm𝒜𝑎𝑥\eta(x):=\sup_{|a|\leq 1}\log\|\mathcal{A}(a,x)\|.
Proposition 3.11.

Let μ𝜇\mu be an ergodic, A𝐴A-invariant Borel probability measure on X𝑋X and suppose ηLd,1(μ)𝜂superscript𝐿𝑑1𝜇\eta\in L^{d,1}(\mu). Then there are

  1. (1)

    an α𝛼\alpha-invariant subset Λ0XsubscriptΛ0𝑋\Lambda_{0}\subset X with μ(Λ0)=1𝜇subscriptΛ01\mu(\Lambda_{0})=1;

  2. (2)

    linear functionals λi:A:subscript𝜆𝑖𝐴\lambda_{i}\colon A\to\mathbb{R} for 1ip1𝑖𝑝1\leq i\leq p;

  3. (3)

    and splittings (x)=i=1pEλi(x)𝑥superscriptsubscriptdirect-sum𝑖1𝑝subscript𝐸subscript𝜆𝑖𝑥\mathcal{E}(x)=\bigoplus_{i=1}^{p}E_{\lambda_{i}}(x) into families of mutually transverse, μ𝜇\mu-measurable subbundles Eλi(x)(x)subscript𝐸subscript𝜆𝑖𝑥𝑥E_{\lambda_{i}}(x)\subset\mathcal{E}(x) defined for xΛ0𝑥subscriptΛ0x\in\Lambda_{0}

such that

  1. (a)

    𝒜(s,x)Eλi(x)=Eλi(sx)𝒜𝑠𝑥subscript𝐸subscript𝜆𝑖𝑥subscript𝐸subscript𝜆𝑖𝑠𝑥\mathcal{A}(s,x)E_{\lambda_{i}}(x)=E_{\lambda_{i}}(s\cdot x) and

  2. (b)

    lim|s|log𝒜(s,x)(v)λi(s)|s|=0subscript𝑠norm𝒜𝑠𝑥𝑣subscript𝜆𝑖𝑠𝑠0\displaystyle\lim_{|s|\to\infty}\frac{\log\|\mathcal{A}(s,x)(v)\|-\lambda_{i}(s)}{|s|}=0

for all xΛ0𝑥subscriptΛ0x\in\Lambda_{0} and all vEλi(p){0}𝑣subscript𝐸subscript𝜆𝑖𝑝0v\in E_{\lambda_{i}}(p)\smallsetminus\{0\}.

Note that (b) implies for vEλi(x)𝑣subscript𝐸subscript𝜆𝑖𝑥v\in E_{\lambda_{i}}(x) the weaker result that for sA𝑠𝐴s\in A,

limk±1klog𝒜(sk,x)(v)=λi(s).subscript𝑘plus-or-minus1𝑘norm𝒜superscript𝑠𝑘𝑥𝑣subscript𝜆𝑖𝑠\lim_{k\to\pm\infty}\tfrac{1}{k}\log\|\mathcal{A}(s^{k},x)(v)\|=\lambda_{i}(s).

Also note that for sA𝑠𝐴s\in A, and μ𝜇\mu an A𝐴A-invariant, A𝐴A-ergodic measure that

λtop,s,μ,𝒜=maxiλi(s).subscript𝜆top𝑠𝜇𝒜subscript𝑖subscript𝜆𝑖𝑠\lambda_{\mathrm{top},s,\mu,\mathcal{A}}=\max_{i}\lambda_{i}(s). (17)

If μ𝜇\mu is not A𝐴A-ergodic, we have the following.

Claim 3.12.

Let μ𝜇\mu be an A𝐴A-invariant measure with ηLd,1(μ)𝜂superscript𝐿𝑑1𝜇\eta\in L^{d,1}(\mu) and λtop,s,μ,𝒜>0subscript𝜆top𝑠𝜇𝒜0\lambda_{\mathrm{top},s,\mu,\mathcal{A}}>0 for some sA𝑠𝐴s\in A. Then there is an A𝐴A-ergodic component μsuperscript𝜇\mu^{\prime} of μ𝜇\mu with

  1. (1)

    ηLd,1(μ)𝜂superscript𝐿𝑑1superscript𝜇\eta\in L^{d,1}(\mu^{\prime});

  2. (2)

    there is non-zero Lyapunov exponent λj0subscript𝜆𝑗0\lambda_{j}\neq 0 for the A𝐴A-action on (X,μ).𝑋superscript𝜇(X,\mu^{\prime}).

We have the following which follows from the above definitions.

Lemma 3.13.

Let μ𝜇\mu be an A𝐴A-invariant probability measure on X𝑋X with exponentially small mass in the cusps. Suppose that 𝒜𝒜\mathcal{A} is a tempered cocycle. Then ηLq(μ)𝜂superscript𝐿𝑞𝜇\eta\in L^{q}(\mu) for all q1𝑞1q\geq 1. In particular, ηLd,1(μ)𝜂superscript𝐿𝑑1𝜇\eta\in L^{d,1}(\mu).

3.8. Applications to the suspension action

We summarize the previous discussion in the setting in which we will apply the above results in the sequel. Recall we work with in a fiber bundle with compact fiber

MMα=(G×M)/Γ𝜋G/Γ𝑀superscript𝑀𝛼𝐺𝑀Γ𝜋𝐺ΓM\rightarrow M^{\alpha}=(G\times M)/\Gamma\xrightarrow{\pi}G/\Gamma

over non-compact base G/Γ𝐺ΓG/\Gamma. From the discussion in [BRHW, Section 2.1], we may equip G×M𝐺𝑀G\times M with a C1superscript𝐶1C^{1} metric that is

  1. (1)

    ΓΓ\Gamma-invariant;

  2. (2)

    the restriction to G𝐺G-orbits coincides with the fixed right-invariant metric on G𝐺G;

  3. (3)

    there is a Siegel fundamental set DG𝐷𝐺D\subset G on which the restrictions to the fibers of the metrics are uniformly comparable.

The metric then descends to a C1superscript𝐶1C^{1} Riemannian metric on Mαsuperscript𝑀𝛼M^{\alpha}. We fix this metric for the remainder. It follows that the diameter of any fiber of Mαsuperscript𝑀𝛼M^{\alpha} is uniformly bounded. It then follows that if μ𝜇\mu is a measure on Mαsuperscript𝑀𝛼M^{\alpha} then the image ν=πμ𝜈subscript𝜋𝜇\nu=\pi_{*}\mu in G/Γ𝐺ΓG/\Gamma has exponentially small mass in the cusps if and only if μ𝜇\mu does; moreover, a family {μζ}subscript𝜇𝜁\{\mu_{\zeta}\} of probability measures on Mαsuperscript𝑀𝛼M^{\alpha} has uniformly exponentially small mass in the cusps if and only if the family of projected measures {πμζ}subscript𝜋subscript𝜇𝜁\{\pi_{*}\mu_{\zeta}\} on G/Γ𝐺ΓG/\Gamma does. Note that by averaging the metric over the left-action of K𝐾K, we may also assume that the metric is left-K𝐾K-invariant. This, in particular, implies the right-invariant metric on G𝐺G in (2)2(2) above is left-K𝐾K-invariant.

For the remainder, the cocycle of interest will be the fiberwise derivative cocycle on the fiberwise tangent bundle,

𝒜(g,x):FF,𝒜(g,x)=Dxg|F.:𝒜𝑔𝑥formulae-sequence𝐹𝐹𝒜𝑔𝑥evaluated-atsubscript𝐷𝑥𝑔𝐹\mathcal{A}(g,x)\colon F\to F,\quad\mathcal{A}(g,x)={D_{x}g}{|_{{F}}}.

Given gG𝑔𝐺g\in G and a g𝑔g-invariant probability measure on Mαsuperscript𝑀𝛼M^{\alpha}, the average leading Lyapunov exponent for the fiberwise derivative cocycle for translation by g𝑔g is written either as λtop,μ,gFsubscriptsuperscript𝜆𝐹top𝜇𝑔\lambda^{F}_{\mathrm{top},\mu,g} or as λtop,μ,g,𝒜subscript𝜆top𝜇𝑔𝒜\lambda_{\mathrm{top},\mu,g,\mathcal{A}}.

The next observation we need is a variant of a fairly standard observation about cocycle over the suspension action.

Lemma 3.14.

The fiberwise derivative cocycle Dxg|Fevaluated-atsubscript𝐷𝑥𝑔𝐹{D_{x}g}{|_{{F}}} is tempered.

Proof.

Write π:MαG/Γ:𝜋superscript𝑀𝛼𝐺Γ\pi\colon M^{\alpha}\to G/\Gamma. By the construction of the metric in the fibers of Mαsuperscript𝑀𝛼M^{\alpha} there is a C>0𝐶0C>0 with the following properties: given xMα𝑥superscript𝑀𝛼x\in M^{\alpha} and gG𝑔𝐺g\in G, writing x¯=π(x)G/Γ¯𝑥𝜋𝑥𝐺Γ\bar{x}=\pi(x)\in G/\Gamma we have

Dxg|FCβ(g,x¯)+1\|{D_{x}g}{|_{{F}}}\|\leq C^{\beta(g,\bar{x})+1}

and

m(Dxg|F)Cβ(g,x¯)1.𝑚evaluated-atsubscript𝐷𝑥𝑔𝐹superscript𝐶𝛽𝑔¯𝑥1m({D_{x}g}{|_{{F}}})\geq C^{-\beta(g,\bar{x})-1}.

The conclusion is then an immediate consequence of Lemma 2.1. ∎

We now assemble the consequences of the results in this section in the form we will use them below in a pair of lemmas. The first is just a special case of Corollary 3.7.

Lemma 3.15.

Let sA𝑠𝐴s\in A and let ν𝜈\nu be an s𝑠s-invariant measure on G/Γ𝐺ΓG/\Gamma with exponentially small mass in the cusps. Let μ𝜇\mu be an s𝑠s-invariant measure on Mαsuperscript𝑀𝛼M^{\alpha} projecting to ν𝜈\nu. Then the average leading Lyapunov exponent for the fiberwise derivative cocycle, λtop,μ,sF,subscriptsuperscript𝜆𝐹top𝜇𝑠\lambda^{F}_{\mathrm{top},\mu,s}, is finite.

The second lemma summarizes the above abstract results in the setting of G𝐺G acting on Mαsuperscript𝑀𝛼M^{\alpha}.

Lemma 3.16.

Let sA𝑠𝐴s\in A and let ν𝜈\nu be an s𝑠s-invariant measure on G/Γ𝐺ΓG/\Gamma with exponentially small mass in the cusps. Let μ𝜇\mu be an s𝑠s-invariant measure on Mαsuperscript𝑀𝛼M^{\alpha} projecting to ν𝜈\nu.

  1. (1)

    For any amenable subgroup HCG(s)𝐻subscript𝐶𝐺𝑠H\subset C_{G}(s), if ν𝜈\nu is H𝐻H-invariant then

    1. (a)

      for any Følner sequence of precompact sets Fnsubscript𝐹𝑛F_{n} in H𝐻H, the family {Fnμ}subscript𝐹𝑛𝜇\{F_{n}\ast\mu\} has uniformly exponentially small mass in the cusps; and

    2. (b)

      for any subsequential limit μsuperscript𝜇\mu^{\prime} of {Fnμ}subscript𝐹𝑛𝜇\{F_{n}\ast\mu\} we have

      λtop,s,μFλtop,s,μF.superscriptsubscript𝜆top𝑠𝜇𝐹superscriptsubscript𝜆top𝑠superscript𝜇𝐹\lambda_{\mathrm{top},s,\mu}^{F}\leq\lambda_{\mathrm{top},s,\mu^{\prime}}^{F}.
  2. (2)

    For any one-parameter unipotent subgroup U𝑈U centralized by s𝑠s

    1. (a)

      the family {UTμ}superscript𝑈𝑇𝜇\{U^{T}\ast\mu\} has uniformly exponentially small mass in the cusps; and

    2. (b)

      for any accumulation point μsuperscript𝜇\mu^{\prime} of {UTμ}superscript𝑈𝑇𝜇\{U^{T}\ast\mu\} as T𝑇T\to\infty we have

      λtop,s,μFλtop,s,μF.superscriptsubscript𝜆top𝑠𝜇𝐹superscriptsubscript𝜆top𝑠superscript𝜇𝐹\lambda_{\mathrm{top},s,\mu}^{F}\leq\lambda_{\mathrm{top},s,\mu^{\prime}}^{F}.
Proof.

Part (a) of the first conclusion is immediate since H𝐻H-invariance of ν𝜈\nu implies ν=π(Fnμ)𝜈subscript𝜋subscript𝐹𝑛𝜇\nu=\pi_{*}(F_{n}\ast\mu) for all n𝑛n; part (b) then follows from Lemma 3.9. The second conclusion follows from Proposition 3.2 and Lemma 3.9. ∎

We remark that we will also use Lemma 3.10 in the proof of the main theorem, but we do not reformulate a special case of it here since the reformulation adds little clarity.

4. Subexponential growth of derivatives for unipotent elements

In this section we show that the restriction of the action α𝛼\alpha to certain unipotent elements in each copy Λi,jSL(2,)subscriptΛ𝑖𝑗SL2\Lambda_{i,j}\cong\mathrm{SL}(2,\mathbb{Z}) have uniform subexponential growth of derivatives with respect to a right-invariant distance on SL(2,)SL2\mathrm{SL}(2,\mathbb{R}). Note that each SL(2,)SL2\mathrm{SL}(2,\mathbb{R}) is geodesically embedded whence the SL(2,)SL2\mathrm{SL}(2,\mathbb{R}) distance is the same as the SL(m,)SL𝑚\mathrm{SL}(m,\mathbb{R}) distance. By [LMR1, LMR2], the SL(m,)SL𝑚\mathrm{SL}(m,\mathbb{R}) distance is quasi-isometric to the word-length in SL(m,)SL𝑚\mathrm{SL}(m,\mathbb{Z}). Recall that d(,)𝑑d(\cdot,\cdot) denotes a right-invariant distance on SL(m,)SL𝑚\mathrm{SL}(m,\mathbb{R}) and that IdId\operatorname{Id} is the identity in SL(m,)SL𝑚\mathrm{SL}(m,\mathbb{R}).

For 1i<jn1𝑖𝑗𝑛1\leq i<j\neq n, let Λi,jSL(2,)subscriptΛ𝑖𝑗SL2\Lambda_{i,j}\cong\mathrm{SL}(2,\mathbb{Z}) be the copy of SL(2,)SL2\mathrm{SL}(2,\mathbb{Z}) in SL(m,)SL𝑚\mathrm{SL}(m,\mathbb{Z}) corresponding to the elements in SL(m,)SL𝑚\mathrm{SL}(m,\mathbb{Z}) which acts only on the lattice 2<msuperscript2superscript𝑚\mathbb{Z}^{2}<\mathbb{Z}^{m} generated by {ei,ej}subscript𝑒𝑖subscript𝑒𝑗\{e_{i},e_{j}\}. Note that as all Λi,jsubscriptΛ𝑖𝑗\Lambda_{i,j} are conjugate under the Weyl group, it suffices to work with one of them.

Define the unipotent element u:=[1101]assign𝑢matrix1101u:=\begin{bmatrix}1&1\\ 0&1\end{bmatrix} viewed as an element of Λi,jsubscriptΛ𝑖𝑗\Lambda_{i,j}. Note that any upper or lower triangular unipotent element of Λi,jsubscriptΛ𝑖𝑗\Lambda_{i,j} is conjugate to a power of u𝑢u under the Weyl group.

Proposition 4.1 (Subexponential growth of derivatives for unipotent elements).

For any Λi,jsubscriptΛ𝑖𝑗\Lambda_{i,j} and any ε>0𝜀0\varepsilon>0, there exists Nε>0subscript𝑁𝜀0N_{\varepsilon}>0 such that for any nNε𝑛subscript𝑁𝜀n\geq N_{\varepsilon}:

D(α(un))eεd(un,Id)norm𝐷𝛼superscript𝑢𝑛superscript𝑒𝜀𝑑superscript𝑢𝑛Id\|D(\alpha(u^{n}))\|\leq e^{\varepsilon d(u^{n},\operatorname{Id})}

To establish Proposition 4.1, we first show that generic elements in SL(2,)SL2\mathrm{SL}(2,\mathbb{Z}) have uniform subexponential growth of derivatives. This first part requires reusing most of the key arguments from [BFH] in a slightly modified form. We encourage the reader to read that paper first.

4.1. Slow growth for “most” elements in SL(2,)SL2\mathrm{SL}(2,\mathbb{Z})

For ε>0𝜀0\varepsilon>0, k>0𝑘0k>0, and xSL(2,)𝑥SL2x\in\mathrm{SL}(2,\mathbb{R}), we make the following definitions:

  1. (1)

    For SSL(2,)𝑆SL2S\subset\mathrm{SL}(2,\mathbb{R}) let |S|𝑆|S| denote the Haar-volume of S𝑆S.

  2. (2)

    Let K=SO(2)SL(2,)𝐾SO2SL2K=\mathrm{SO}(2)\subset\mathrm{SL}(2,\mathbb{R}). For SK𝑆𝐾S\subset K let |S|𝑆|S| denote the Haar-volume of S𝑆S.

  3. (3)

    Let Bk(x)subscript𝐵𝑘𝑥B_{k}(x) denote the ball of radius k𝑘k centered at x𝑥x in SL(2,)SL2\mathrm{SL}(2,\mathbb{R}).

  4. (4)

    Let Tk:=Bk(Id)SL(2,)assignsubscript𝑇𝑘subscript𝐵𝑘IdSL2T_{k}:=B_{k}(\operatorname{Id})\cap\mathrm{SL}(2,\mathbb{Z}). Given SSL(2,)𝑆SL2S\subset\mathrm{SL}(2,\mathbb{Z}) write |S|𝑆|S| for the cardinality of S𝑆S.

  5. (5)

    Define the set of ε𝜀\varepsilon-bad elements to be

    Mε,k:={γTk such that D(α(γ))eεk}.assignsubscript𝑀𝜀𝑘𝛾subscript𝑇𝑘 such that norm𝐷𝛼𝛾superscript𝑒𝜀𝑘M_{\varepsilon,k}:=\{\gamma\in T_{k}\text{ such that }\|D(\alpha(\gamma))\|\geq e^{\varepsilon k}\}.
  6. (6)

    Define the set of ε𝜀\varepsilon-good elements to be

    Gε,k:=TkMε,k.assignsubscript𝐺𝜀𝑘subscript𝑇𝑘subscript𝑀𝜀𝑘G_{\varepsilon,k}:=T_{k}\setminus M_{\varepsilon,k}.

To establish Proposition 4.1, we first show that the set Gε,ksubscript𝐺𝜀𝑘G_{\varepsilon,k} contains a positive proportion of Tksubscript𝑇𝑘T_{k} when k𝑘k is large enough.

Proposition 4.2.

For any δ>0𝛿0\delta>0, the set Gε,ksubscript𝐺𝜀𝑘G_{\varepsilon,k} has at least (1δ)|Tk|1𝛿subscript𝑇𝑘(1-\delta)|T_{k}| elements for every sufficiently large k𝑘k.

We have the following well-known fact. See for instance [EM1, Section 2].

Lemma 4.3.

There exist positive constants c,C𝑐𝐶c,C such that for any k0𝑘0k\geq 0:

c|Bk||Tk|C|Bk|.𝑐subscript𝐵𝑘subscript𝑇𝑘𝐶subscript𝐵𝑘c|B_{k}|\leq|T_{k}|\leq C|B_{k}|.

For an element xSL(2,)𝑥SL2x\in\mathrm{SL}(2,\mathbb{R}), let x¯¯𝑥\bar{x} denote the projection in SL(2,)/SL(2,)SL2SL2\mathrm{SL}(2,\mathbb{R})/\mathrm{SL}(2,\mathbb{Z}). Define

Dx¯gFiber=sup{Dyg|F:yMα,π(y)=x¯}.\|D_{\bar{x}}g\|_{\text{Fiber}}=\sup\{\|{D_{y}g}{|_{{F}}}\|:y\in M^{\alpha},\pi(y)=\bar{x}\}.

Let

Gε,k(x):={gBk(x) such that Dx¯gFibereεd(g,Id)}.assignsubscriptsuperscript𝐺𝜀𝑘𝑥conditional-set𝑔subscript𝐵𝑘𝑥 such that evaluated-atsubscript𝐷¯𝑥𝑔Fibersuperscript𝑒𝜀𝑑𝑔IdG^{\prime}_{\varepsilon,k}(x):=\{g\in B_{k}(x)\text{ such that }\|D_{\bar{x}}g\|_{\text{Fiber}}\leq e^{\varepsilon d(g,\operatorname{Id})}\}.
Lemma 4.4.

For almost every xSL(2,)𝑥SL2x\in\mathrm{SL}(2,\mathbb{R}) and any δ>0𝛿0\delta>0 we have

|Gε,k(x)|>(1δ)|Bk|subscriptsuperscript𝐺𝜀𝑘𝑥1𝛿subscript𝐵𝑘|G^{\prime}_{\varepsilon,k}(x)|>(1-\delta)|B_{k}|

for all k𝑘k sufficiently large.

Proof.

Let atSL(2,)superscript𝑎𝑡SL2a^{t}\in\mathrm{SL}(2,\mathbb{R}) be the matrix

at:=[et00et].assignsuperscript𝑎𝑡matrixsuperscript𝑒𝑡00superscript𝑒𝑡a^{t}:=\begin{bmatrix}e^{t}&0\\ 0&e^{-t}\end{bmatrix}.

Recall that the action of the one-parameter diagonal subgroup {at}superscript𝑎𝑡\{a^{t}\} on SL(2,)/SL(2,)SL2SL2\mathrm{SL}(2,\mathbb{R})/\mathrm{SL}(2,\mathbb{Z}) is ergodic with respect to Haar measure.

Let \mathcal{M} denote the set of Borel probability measures on SL(2,)/SL(2,)SL2SL2\mathrm{SL}(2,\mathbb{R})/\mathrm{SL}(2,\mathbb{Z}) equipped with the standard topology (dual to bounded continuous functions). The topology on \mathcal{M} is metrizable (see [Bil, Theorem 6.8]); fix a metric on ρsubscript𝜌\rho_{\mathcal{M}} on \mathcal{M}.

Consider the function ψ:SL(2,)/SL(2,):𝜓SL2SL2\psi\colon\mathrm{SL}(2,\mathbb{R})/\mathrm{SL}(2,\mathbb{Z})\to\mathbb{R} given by ψ(x):=eηd(x,x0)assign𝜓𝑥superscript𝑒𝜂𝑑𝑥subscript𝑥0\psi(x):=e^{\eta d(x,x_{0})} where x0=SL(2,)subscript𝑥0SL2x_{0}=\mathrm{SL}(2,\mathbb{Z}) is the identity coset and η>0𝜂0\eta>0 is chosen sufficiently small so that ψ𝜓\psi is L1superscript𝐿1L^{1} with respect to the Haar measure. By the pointwise ergodic theorem, for almost every xSL(2,)𝑥SL2x\in\mathrm{SL}(2,\mathbb{R}) and almost every k1SO(2)subscript𝑘1SO2k_{1}\in\mathrm{SO}(2) we have

limT1T0Tψ(atk1x¯)𝑑t=SL(2,)/SL(2,)ψ𝑑Haar<.subscript𝑇1𝑇superscriptsubscript0𝑇𝜓superscript𝑎𝑡subscript𝑘1¯𝑥differential-d𝑡subscriptSL2SL2𝜓differential-dHaar\lim_{T\to\infty}\frac{1}{T}\int_{0}^{T}\psi(a^{t}k_{1}\bar{x})\ dt=\int_{\mathrm{SL}(2,\mathbb{R})/\mathrm{SL}(2,\mathbb{Z})}\psi\ d\text{Haar}<\infty. (18)

Similarly, for almost every xSL(2,)𝑥SL2x\in\mathrm{SL}(2,\mathbb{R}) and almost every k1SO(2)subscript𝑘1SO2k_{1}\in\mathrm{SO}(2) we have

limT1T0Tδatk1x¯𝑑t=Haar.subscript𝑇1𝑇superscriptsubscript0𝑇subscript𝛿superscript𝑎𝑡subscript𝑘1¯𝑥differential-d𝑡Haar\lim_{T\to\infty}\frac{1}{T}\int_{0}^{T}\delta_{a^{t}k_{1}\bar{x}}\ dt=\text{Haar}. (19)

Let SSL(2,)𝑆SL2S\subset\mathrm{SL}(2,\mathbb{R}) be the set of xSL(2,)𝑥SL2x\in\mathrm{SL}(2,\mathbb{R}) such that (18) and (19) hold for almost every k1SO(2)subscript𝑘1SO2k_{1}\in\mathrm{SO}(2). The set S𝑆S is SL(2,)SL2\mathrm{SL}(2,\mathbb{Z})-invariant and co-null. We show any xS𝑥𝑆x\in S satisfies the conclusion of the lemma.

For fixed xS𝑥𝑆x\in S and fixed δ>0𝛿0\delta>0, there exist Tδ=Tδ(x)subscript𝑇𝛿subscript𝑇𝛿𝑥T_{\delta}=T_{\delta}(x), a sequence Tj=Tj(x)subscript𝑇𝑗subscript𝑇𝑗𝑥T_{j}=T_{j}(x) for j𝑗j\in\mathbb{N}, and a set Kδ=Kδ(x)SO(2)subscript𝐾𝛿subscript𝐾𝛿𝑥SO2K_{\delta}=K_{\delta}(x)\subset\mathrm{SO}(2) such that |Kδ|(1δ/2)|SO(2)|subscript𝐾𝛿1𝛿2SO2|K_{\delta}|\geq(1-\delta/2)|\mathrm{SO}(2)| with the property that for any k1Kδsubscript𝑘1subscript𝐾𝛿k_{1}\in K_{\delta} and any TTδ𝑇subscript𝑇𝛿T\geq T_{\delta} we have

1T0Tψ(atk1x¯)𝑑t<2SL(2,)/SL(2,)ψ𝑑Haar1𝑇superscriptsubscript0𝑇𝜓superscript𝑎𝑡subscript𝑘1¯𝑥differential-d𝑡2subscriptSL2SL2𝜓differential-dHaar\frac{1}{T}\int_{0}^{T}\psi(a^{t}k_{1}\bar{x})\ dt<2\int_{\mathrm{SL}(2,\mathbb{R})/\mathrm{SL}(2,\mathbb{Z})}\psi\ d\text{Haar} (20)

and for each 1j1𝑗1\leq j

ρ(1T0Tδatk1x¯𝑑t,Haar)<1j.subscript𝜌1𝑇superscriptsubscript0𝑇subscript𝛿superscript𝑎𝑡subscript𝑘1¯𝑥differential-d𝑡Haar1𝑗\rho_{\mathcal{M}}\left(\frac{1}{T}\int_{0}^{T}\delta_{a^{t}k_{1}\bar{x}}\ dt,\text{Haar}\right)<\frac{1}{j}. (21)

for all TTj𝑇subscript𝑇𝑗T\geq T_{j}. To finish the proof of the lemma, define the set

Gk′′(x):={k1atk2 where k1SO(2),k2Kδ(x) and (δ/2)k<t<k}.assignsubscriptsuperscript𝐺′′𝑘𝑥formulae-sequencesubscript𝑘1superscript𝑎𝑡subscript𝑘2 where subscript𝑘1SO2subscript𝑘2subscript𝐾𝛿𝑥 and 𝛿2𝑘𝑡𝑘G^{\prime\prime}_{k}(x):=\{k_{1}a^{t}k_{2}\text{ where }k_{1}\in\mathrm{SO}(2),k_{2}\in K_{\delta}(x)\text{ and }(\delta/2)k<t<k\}.

For k𝑘k large enough, we have that |Gk′′(x)|(1δ)|Bk|subscriptsuperscript𝐺′′𝑘𝑥1𝛿subscript𝐵𝑘|G^{\prime\prime}_{k}(x)|\geq(1-\delta)|B_{k}|. We claim that

Gk′′(x)Gε,k(x)subscriptsuperscript𝐺′′𝑘𝑥subscriptsuperscript𝐺𝜀𝑘𝑥G^{\prime\prime}_{k}(x)\subset G^{\prime}_{\varepsilon,k}(x) (22)

for k𝑘k sufficiently large. For the sake of contradiction, suppose (22) fails. Using that the norm on F𝐹F is chosen to be K𝐾K-invariant, there exists xnSL(2,)subscript𝑥𝑛SL2x_{n}\in\mathrm{SL}(2,\mathbb{R}) with each xnsubscript𝑥𝑛x_{n} in the Kδ(x)subscript𝐾𝛿𝑥K_{\delta}(x)-orbit of x𝑥x such that Dxn(atn)Fibereεtnsubscriptnormsubscript𝐷subscript𝑥𝑛superscript𝑎subscript𝑡𝑛Fibersuperscript𝑒𝜀subscript𝑡𝑛\|D_{x_{n}}(a^{t_{n}})\|_{\text{Fiber}}\geq e^{\varepsilon t_{n}} for some sequence tnsubscript𝑡𝑛t_{n}\to\infty. Moreover, the corresponding empirical measures

ηn:=1tn0tnδatx¯n𝑑tassignsubscript𝜂𝑛1subscript𝑡𝑛superscriptsubscript0subscript𝑡𝑛subscript𝛿superscript𝑎𝑡subscript¯𝑥𝑛differential-d𝑡\eta_{n}:=\frac{1}{t_{n}}\int_{0}^{t_{n}}\delta_{a^{t}\bar{x}_{n}}\ dt

have uniformly exponentially small mass in the cusps by equation (20).

By Lemma 3.10 and (21), a subsequence of the measures ηnsubscript𝜂𝑛\eta_{n} converge to an atsuperscript𝑎𝑡a^{t}-invariant measure μ0subscript𝜇0\mu_{0} on Mαsuperscript𝑀𝛼M^{\alpha} whose projection to SL(m,)/SL(m,)SL𝑚SL𝑚\mathrm{SL}(m,\mathbb{R})/\mathrm{SL}(m,\mathbb{Z}) is Haar measure on the embedded modular surface SL(2,)/SL(2,)SL2SL2\mathrm{SL}(2,\mathbb{R})/\mathrm{SL}(2,\mathbb{Z}) and has positive fiberwise Lyapunov exponent for the action of a1superscript𝑎1a^{1}. Since atsuperscript𝑎𝑡a^{t} is ergodic on SL(2,)/SL(2,)SL2SL2\mathrm{SL}(2,\mathbb{R})/\mathrm{SL}(2,\mathbb{Z}), we can assume μ0subscript𝜇0\mu_{0} is ergodic by taking an ergodic component without changing any other properties.

We average as in [BFH] to improve μ0subscript𝜇0\mu_{0} to a measure whose projection is the Haar measure on SL(m,)/SL(m,)SL𝑚SL𝑚\mathrm{SL}(m,\mathbb{R})/\mathrm{SL}(m,\mathbb{Z}). Difficulties related to escape of mass are handled by the preliminaries in Section 3.

As above, we note that there is a canonical copy of H2=SL(m2,)subscript𝐻2SL𝑚2H_{2}=\mathrm{SL}(m-2,\mathbb{R}) in SL(m,)SL𝑚\mathrm{SL}(m,\mathbb{R}) commuting with our chosen H1=SL(2,)subscript𝐻1SL2H_{1}=\mathrm{SL}(2,\mathbb{R}). Recall A𝐴A is the Cartan subgroup of SL(m,)SL𝑚\mathrm{SL}(m,\mathbb{R}) of positive diagonal matrices. The subgroup A𝐴A contains the one-parameter group {at}superscript𝑎𝑡\{a^{t}\} and a Cartan subgroup of H2subscript𝐻2H_{2}. Let

  • A1=AH1={at}subscript𝐴1𝐴subscript𝐻1superscript𝑎𝑡A_{1}=A\cap H_{1}=\{a^{t}\},

  • A2=AH2subscript𝐴2𝐴subscript𝐻2A_{2}=A\cap H_{2}, and

  • A=AH1×H2superscript𝐴𝐴subscript𝐻1subscript𝐻2A^{\prime}=A\cap H_{1}\times H_{2}.

Note that A<Asuperscript𝐴𝐴A^{\prime}<A has codimension one. Our chosen modular surface SL(2,)/SL(2,)SL(m,)/SL(m,)SL2SL2SL𝑚SL𝑚\mathrm{SL}(2,\mathbb{R})/\mathrm{SL}(2,\mathbb{Z})\subset\mathrm{SL}(m,\mathbb{R})/\mathrm{SL}(m,\mathbb{Z}) is such that

SL(2,)/SL(2,)SL2SL2\displaystyle\mathrm{SL}(2,\mathbb{R})/\mathrm{SL}(2,\mathbb{Z}) SL(2,)/SL(2,)×SL(m2,)/SL(m2,)absentSL2SL2SL𝑚2SL𝑚2\displaystyle\subset\mathrm{SL}(2,\mathbb{R})/\mathrm{SL}(2,\mathbb{Z})\times\mathrm{SL}(m-2,\mathbb{R})/\mathrm{SL}(m-2,\mathbb{Z})
SL(m,)/SL(m,).absentSL𝑚SL𝑚\displaystyle\subset\mathrm{SL}(m,\mathbb{R})/\mathrm{SL}(m,\mathbb{Z}).

Define an Asuperscript𝐴A^{\prime}-ergodic, Asuperscript𝐴A^{\prime}-invariant measure μ1subscript𝜇1\mu_{1} on Mαsuperscript𝑀𝛼M^{\alpha} that projects to Haar measure on SL(2,)/SL(2,)×SL(m2,)/SL(m2,)SL2SL2SL𝑚2SL𝑚2\mathrm{SL}(2,\mathbb{R})/\mathrm{SL}(2,\mathbb{Z})\times\mathrm{SL}(m-2,\mathbb{R})/\mathrm{SL}(m-2,\mathbb{Z}) as follows: Let M2,m2αsubscriptsuperscript𝑀𝛼2𝑚2M^{\alpha}_{2,m-2} denote the restriction of the fiber-bundle Mαsuperscript𝑀𝛼M^{\alpha} to SL(2,)/SL(2,)×SL(m2,)/SL(m2,)SL2SL2SL𝑚2SL𝑚2\mathrm{SL}(2,\mathbb{R})/\mathrm{SL}(2,\mathbb{Z})\times\mathrm{SL}(m-2,\mathbb{R})/\mathrm{SL}(m-2,\mathbb{Z}). Pick point y𝑦y in SL(m2,)/SL(m2,)SL𝑚2SL𝑚2\mathrm{SL}(m-2,\mathbb{R})/\mathrm{SL}(m-2,\mathbb{Z}) that equidistributes to the Haar measure on SL(m2,)/SL(m2,)SL𝑚2SL𝑚2\mathrm{SL}(m-2,\mathbb{R})/\mathrm{SL}(m-2,\mathbb{Z}) under a Følner sequence in A2subscript𝐴2A_{2}. Consider μ0subscript𝜇0\mu_{0} as a measure on the restriction of Mαsuperscript𝑀𝛼M^{\alpha} to SL(2,)/SL(2,)×{y}SL2SL2𝑦\mathrm{SL}(2,\mathbb{R})/\mathrm{SL}(2,\mathbb{Z})\times\{y\}. Now average μ0subscript𝜇0\mu_{0} over a Følner sequence in A2subscript𝐴2A_{2} and take a limit μ^1subscript^𝜇1\hat{\mu}_{1}. Note that μ^1subscript^𝜇1\hat{\mu}_{1} has positive fiberwise Lyapunov exponent λtop,a1,μ1F>0superscriptsubscript𝜆topsuperscript𝑎1subscript𝜇1𝐹0\lambda_{\mathrm{top},a^{1},\mu_{1}}^{F}>0. This can be seen by mimicking the proof of Lemma 3.9. Let μ1subscript𝜇1\mu_{1} be an Asuperscript𝐴A^{\prime} ergodic component of μ^1subscript^𝜇1\hat{\mu}_{1}, then the measure μ1subscript𝜇1\mu_{1} has the desired properties and is supported on the subset of Mαsuperscript𝑀𝛼M^{\alpha} defined by restricting the bundle to SL(2,)/SL(2,)×SL(m2,)/SL(m2,)SL2SL2SL𝑚2SL𝑚2\mathrm{SL}(2,\mathbb{R})/\mathrm{SL}(2,\mathbb{Z})\times\mathrm{SL}(m-2,\mathbb{R})/\mathrm{SL}(m-2,\mathbb{Z}).

We consider the Asuperscript𝐴A^{\prime}-action on (Mα,μ1)superscript𝑀𝛼subscript𝜇1(M^{\alpha},\mu_{1}) and the fiberwise derivative cocycle 𝒜(g,y)=Dyg|F𝒜𝑔𝑦evaluated-atsubscript𝐷𝑦𝑔𝐹\mathcal{A}(g,y)={D_{y}g}{|_{{F}}}. By (17), there is a non-zero Lyapunov exponent λμ1,AF:A:superscriptsubscript𝜆subscript𝜇1superscript𝐴𝐹superscript𝐴\lambda_{\mu_{1},A^{\prime}}^{F}\colon A^{\prime}\to\mathbb{R} for this action. We apply the averaging procedure in Proposition 3.5 to this measure. Take βsuperscript𝛽\beta^{\prime} to be either α2subscript𝛼2\alpha_{2} or δ𝛿\delta so that β:A:superscript𝛽superscript𝐴\beta^{\prime}\colon A^{\prime}\to\mathbb{R} is not proportional to λμ1,AFsuperscriptsubscript𝜆subscript𝜇1superscript𝐴𝐹\lambda_{\mu_{1},A^{\prime}}^{F}. Choose a0Asubscript𝑎0superscript𝐴a_{0}\in A^{\prime} such that a0ker(β)subscript𝑎0kernelsuperscript𝛽a_{0}\in\ker(\beta^{\prime}) and λμ1,AF(a0)>0superscriptsubscript𝜆subscript𝜇1superscript𝐴𝐹subscript𝑎00\lambda_{\mu_{1},A^{\prime}}^{F}(a_{0})>0. Let U=Uβ𝑈superscript𝑈superscript𝛽U=U^{\beta^{\prime}} and let μ2subscript𝜇2\mu_{2} be any subsequential limit of UTμ1superscript𝑈𝑇subscript𝜇1U^{T}\ast\mu_{1} as T𝑇T\rightarrow\infty. Then μ2subscript𝜇2\mu_{2} is a0subscript𝑎0a_{0}-invariant, and has positive fiberwise Lyapunov exponent λtop,a0,μ2F>0superscriptsubscript𝜆topsubscript𝑎0subscript𝜇2𝐹0\lambda_{\mathrm{top},a_{0},\mu_{2}}^{F}>0. Moreover, πμ2subscript𝜋subscript𝜇2\pi_{*}\mu_{2} is H𝐻H-invariant. By Lemma 3.16 and Proposition 3.5, μ2subscript𝜇2\mu_{2} has exponentially small mass in the cusps. We may also assume μ2subscript𝜇2\mu_{2} is ergodic by passing to an ergodic component and by Claim 3.12 assume μ2subscript𝜇2\mu_{2} has a non-zero fiberwise Lyapunov exponent λμ2,AFsuperscriptsubscript𝜆subscript𝜇2superscript𝐴𝐹\lambda_{\mu_{2},A^{\prime}}^{F} for the Asuperscript𝐴A^{\prime}-action.

We now average μ2subscript𝜇2\mu_{2} over Asuperscript𝐴A^{\prime} to obtain μ3subscript𝜇3\mu_{3}. Then μ3subscript𝜇3\mu_{3} has a non-zero fiberwise Lyapunov exponent λμ3,AFsuperscriptsubscript𝜆subscript𝜇3superscript𝐴𝐹\lambda_{\mu_{3},A^{\prime}}^{F} and has exponentially small mass in the cusps by Lemma 3.16(1). Since πμ2subscript𝜋subscript𝜇2\pi_{*}\mu_{2} was Asuperscript𝐴A^{\prime}-invariant, we have πμ2=πμ3subscript𝜋subscript𝜇2subscript𝜋subscript𝜇3\pi_{*}\mu_{2}=\pi_{*}\mu_{3}. Once again, we may pass to an Asuperscript𝐴A^{\prime}-ergodic component of μ3subscript𝜇3\mu_{3} that retains the desired properties.

Take β^^𝛽\hat{\beta} to be either α2subscript𝛼2-\alpha_{2} or δ𝛿-\delta so that β^^𝛽\hat{\beta} is not proportional to λμ3,AFsuperscriptsubscript𝜆subscript𝜇3superscript𝐴𝐹\lambda_{\mu_{3},A^{\prime}}^{F} on Asuperscript𝐴A^{\prime}. Select a1subscript𝑎1a_{1} with λμ3,AF(a1)>0superscriptsubscript𝜆subscript𝜇3superscript𝐴𝐹subscript𝑎10\lambda_{\mu_{3},A^{\prime}}^{F}(a_{1})>0 and β^(a1)=0^𝛽subscript𝑎10\hat{\beta}(a_{1})=0. By Proposition 3.5 and Lemma 3.16, we obtain a new measure μ4subscript𝜇4\mu_{4} with πμ4subscript𝜋subscript𝜇4\pi_{*}\mu_{4} the Haar measure on SL(m,)/SL(m,Z)SL𝑚SL𝑚𝑍\mathrm{SL}(m,\mathbb{R})/\mathrm{SL}(m,Z). We have λtop,a1,μ4F>0superscriptsubscript𝜆topsubscript𝑎1subscript𝜇4𝐹0\lambda_{\mathrm{top},a_{1},\mu_{4}}^{F}>0. Finally, average μ4subscript𝜇4\mu_{4} over all of A𝐴A to obtain μ5subscript𝜇5\mu_{5}. Since πμ4subscript𝜋subscript𝜇4\pi_{*}\mu_{4} is the Haar measure and thus A𝐴A-invariant, we have that πμ4=πμ5subscript𝜋subscript𝜇4subscript𝜋subscript𝜇5\pi_{*}\mu_{4}=\pi_{*}\mu_{5}. By Lemma 3.16, μ5subscript𝜇5\mu_{5} has a non-zero fiberwise Lyapunov exponent λμ5,AFsuperscriptsubscript𝜆subscript𝜇5𝐴𝐹\lambda_{\mu_{5},A}^{F} for the action of A𝐴A. Replace μ5subscript𝜇5\mu_{5} by an ergodic component with positive fiberwise Lyapunov exponent.

Exactly as in [BFH, Section 5.5], we apply [BRHW, Proposition 5.1] and conclude that μ5subscript𝜇5\mu_{5} is a G𝐺G-invariant measure on Mαsuperscript𝑀𝛼M^{\alpha}. We then obtain a contradiction with Zimmer’s cocycle superrigidity theorem. To conclude that μ5subscript𝜇5\mu_{5} is a G𝐺G-invariant, note that [BRHW, Proposition 5.1] holds for actions induced from actions of any lattice in SL(m,)SL𝑚\mathrm{SL}(m,\mathbb{R}) and shows that μ5subscript𝜇5\mu_{5} is invariant under root subgroups corresponding to non-resonant roots. Dimension counting exactly as in [BFH, Section 5.5] shows that the non-resonant roots of SL(m,)SL𝑚\mathrm{SL}(m,\mathbb{R}) generate all of G𝐺G if the dimension of M𝑀M is at most m2𝑚2m-2 or if the dimension of M𝑀M is m1𝑚1m-1 and the action is preserves a volume. ∎

We derive Proposition 4.2 from Lemma 4.4.

Proof of Proposition 4.2.

Fix 0<c<10𝑐10<c<1 sufficiently small so that if d(Id,g)<c𝑑Id𝑔𝑐d(\operatorname{Id},g)<c then DΓgFibereε/4subscriptnormsubscript𝐷Γ𝑔Fibersuperscript𝑒𝜀4\|D_{\Gamma}g\|_{\text{Fiber}}\leq e^{\varepsilon/4}. Fix a point xSL(2,)𝑥SL2x\in\mathrm{SL}(2,\mathbb{R}) as in Lemma 4.4 with d(Id,x)<c𝑑Id𝑥𝑐d(\operatorname{Id},x)<c. Observe that if k1𝑘1k\geq 1 and gGε/4,k(x)𝑔subscriptsuperscript𝐺𝜀4𝑘𝑥g\in G^{\prime}_{\varepsilon/4,k}(x), then gxGε/2,k+c(Id)𝑔𝑥subscriptsuperscript𝐺𝜀2𝑘𝑐Idgx\in G^{\prime}_{\varepsilon/2,k+c}(\operatorname{Id}). In particular, for any δ>0𝛿0\delta>0 we have for all k𝑘k sufficiently large that

|Bk+cGε/2,k+c(Id)|<δC^|Bk|subscript𝐵𝑘𝑐subscriptsuperscript𝐺𝜀2𝑘𝑐Id𝛿^𝐶subscript𝐵𝑘|B_{k+c}\smallsetminus G^{\prime}_{\varepsilon/2,k+c}(\operatorname{Id})|<\delta\hat{C}|B_{k}| (23)

where C^^𝐶\hat{C} is a constant depending on c𝑐c.

Take U𝑈U to be the ball of radius c𝑐c centered at the identity coset in SL(2,)/SL(2,)SL2SL2\mathrm{SL}(2,\mathbb{R})/\mathrm{SL}(2,\mathbb{Z}) and consider lifts of U𝑈U to SL(2,)SL2\mathrm{SL}(2,\mathbb{R}) intersecting the ball Bksubscript𝐵𝑘B_{k}. If a lift of U𝑈U intersects Gε/2,k+c(Id)subscriptsuperscript𝐺𝜀2𝑘𝑐IdG^{\prime}_{\varepsilon/2,k+c}(\operatorname{Id}), then the corresponding element of the deck group SL(2,)SL2\mathrm{SL}(2,\mathbb{Z}) belongs to G3ε/4,k(Id)subscriptsuperscript𝐺3𝜀4𝑘IdG^{\prime}_{3\varepsilon/4,k}(\mathrm{Id}).

Let U~~𝑈\tilde{U} be the set of lifts of U𝑈U. From Lemma 4.3 and (23), it follows that ratio of the measure of U~BkGε/2,k(Id)~𝑈subscript𝐵𝑘subscriptsuperscript𝐺𝜀2𝑘Id\tilde{U}\cap B_{k}\cap G^{\prime}_{\varepsilon/2,k}(\operatorname{Id}) to the measure of U~Bk~𝑈subscript𝐵𝑘\tilde{U}\cap B_{k} goes to one as k𝑘k\to\infty. Finally, since the norms on the fiber of Mαsuperscript𝑀𝛼M^{\alpha} above the identity coset and the original norm on M𝑀M are uniformly comparable, the result follows. ∎

Remark 4.5.

Using large deviations, one can make δ𝛿\delta to be decreasing with k𝑘k, roughly as δk=ek1/1000subscript𝛿𝑘superscript𝑒superscript𝑘11000\delta_{k}=e^{-k^{1/1000}}. See [Ath, EM2]. This is not necessary for our argument.

4.2. Subexponential growth of derivatives for unipotent elements in SL(2,)SL2\mathrm{SL}(2,\mathbb{Z})

We work here with a specific copy of the group SL(2,)2left-normal-factor-semidirect-productSL2superscript2\mathrm{SL}(2,\mathbb{R})\ltimes\mathbb{R}^{2} embedded in SL(m,)SL𝑚\mathrm{SL}(m,\mathbb{R}) and its intersection with the lattice ΓΓ\Gamma; the copy of SL(2,)2left-normal-factor-semidirect-productSL2superscript2\mathrm{SL}(2,\mathbb{R})\ltimes\mathbb{R}^{2} corresponds to the elements of SLm()subscriptSL𝑚\mathrm{SL}_{m}(\mathbb{R}) which differ from the identity matrix only in the first two rows and first three columns. Any unipotent element of any Λi,jΓsubscriptΛ𝑖𝑗Γ\Lambda_{i,j}\subset\Gamma considered in the statement of Proposition 4.1 is conjugate by an element of the Weyl group to a power of the elementary matrix E1,3subscript𝐸13E_{1,3}. Thus, after conjugation, any such element is contained in the distinguished copy of SL(2,)2left-normal-factor-semidirect-productSL2superscript2\mathrm{SL}(2,\mathbb{Z})\ltimes\mathbb{Z}^{2} generated by SL(2,)=Λ1,2SL2subscriptΛ12\mathrm{SL}(2,\mathbb{Z})=\Lambda_{1,2} and the normal subgroup 2superscript2\mathbb{Z}^{2} generated by E1,3subscript𝐸13E_{1,3} and E2,3subscript𝐸23E_{2,3}.

For the reminder of this subsection, we work with this fixed group. Identify H1,2subscript𝐻12H_{1,2} with SL(2,)SL2\mathrm{SL}(2,\mathbb{R}). Let U1,2:={ua,b}assignsubscript𝑈12subscript𝑢𝑎𝑏U_{1,2}:=\{u_{a,b}\} denote the abelian subgroup of SL(m,)SL𝑚\mathrm{SL}(m,\mathbb{R}) consisting of unipotent elements of the form

ua,b:=(10a01b0011)assignsubscript𝑢𝑎𝑏10𝑎missing-subexpressionmissing-subexpression01𝑏missing-subexpressionmissing-subexpression001missing-subexpressionmissing-subexpressionmissing-subexpressionmissing-subexpressionmissing-subexpressionmissing-subexpressionmissing-subexpressionmissing-subexpressionmissing-subexpressionmissing-subexpression1u_{a,b}:=\left(\begin{array}[]{ccccc}1&0&a&&\\ 0&1&b&&\\ 0&0&1&&\\ &&&\ddots&\\ &&&&1\end{array}\right)

Clearly, U1,2subscript𝑈12U_{1,2} is normalized by H1,2subscript𝐻12H_{1,2} and H1,2U1,2SL(2,)2left-normal-factor-semidirect-productsubscript𝐻12subscript𝑈12left-normal-factor-semidirect-productSL2superscript2H_{1,2}\ltimes U_{1,2}\cong\mathrm{SL}(2,\mathbb{R})\ltimes\mathbb{R}^{2}. We have an embedding

SL(2,)2/SL(2,)2SL(m,)/SL(m,)left-normal-factor-semidirect-productleft-normal-factor-semidirect-productSL2superscript2SL2superscript2SL𝑚SL𝑚\mathrm{SL}(2,\mathbb{R})\ltimes\mathbb{R}^{2}/\mathrm{SL}(2,\mathbb{Z})\ltimes\mathbb{Z}^{2}\to\mathrm{SL}(m,\mathbb{R})/\mathrm{SL}(m,\mathbb{Z})

where 2superscript2\mathbb{Z}^{2} is identified with the subgroup generated by the unipotent elements u1,0subscript𝑢10u_{1,0} and u0,1subscript𝑢01u_{0,1}. Note that SL(2,)2/SL(2,)2left-normal-factor-semidirect-productleft-normal-factor-semidirect-productSL2superscript2SL2superscript2\mathrm{SL}(2,\mathbb{R})\ltimes\mathbb{R}^{2}/\mathrm{SL}(2,\mathbb{Z})\ \ltimes\mathbb{Z}^{2} is a torus bundle over the unit-tangent bundle of the modular surface.

Equip 2superscript2\mathbb{Z}^{2} with the Lsubscript𝐿L_{\infty} norm with respect to the generating set {u1,0,u0,1}subscript𝑢10subscript𝑢01\{u_{1,0},u_{0,1}\} and let Bn(2)subscript𝐵𝑛superscript2B_{n}(\mathbb{Z}^{2}) denote the closed ball of radius n𝑛n in 2superscript2\mathbb{Z}^{2} centered at 00 with respect to this norm. Given S2𝑆superscript2S\subset\mathbb{Z}^{2} let |S|𝑆|S| denote the cardinality of the set S𝑆S.

Define the set of “ε𝜀\varepsilon-good unipotent elements” of 2Γsuperscript2Γ\mathbb{Z}^{2}\subset\Gamma, denoted by GUε,n𝐺subscript𝑈𝜀𝑛GU_{\varepsilon,n}, to be the following subset of 2superscript2\mathbb{Z}^{2}:

GUε,n:={ua,bBn(2) such that D(α(ua,b±1))eεlog(n)}.assign𝐺subscript𝑈𝜀𝑛subscript𝑢𝑎𝑏subscript𝐵𝑛superscript2 such that norm𝐷𝛼superscriptsubscript𝑢𝑎𝑏plus-or-minus1superscript𝑒𝜀𝑛GU_{\varepsilon,n}:=\left\{u_{a,b}\in B_{n}(\mathbb{Z}^{2})\text{ such that }\|D(\alpha(u_{a,b}^{\pm 1}))\|\leq e^{\varepsilon\log(n)}\right\}. (24)

The main results of this subsection is the following.

Proposition 4.6.

For any ε>0𝜀0\varepsilon>0, there exists Nε>0subscript𝑁𝜀0N_{\varepsilon}>0 such that if nNε𝑛subscript𝑁𝜀n\geq N_{\varepsilon}, then GUε,n=Bn(2)𝐺subscript𝑈𝜀𝑛subscript𝐵𝑛superscript2GU_{\varepsilon,n}=B_{n}(\mathbb{Z}^{2})

Proposition 4.1 follows from Proposition 4.6 using that any subgroup undelimited-⟨⟩superscript𝑢𝑛\langle u^{n}\rangle in Proposition 4.1 is conjugate to a subgroup of the group 2superscript2\mathbb{Z}^{2} and the fact that d(un,Id)=O(log(n))𝑑superscript𝑢𝑛Id𝑂𝑛d(u^{n},\operatorname{Id})=O(\log(n)) from (6). The proof of Proposition 4.6 consists of conjugating elements of U1,2subscript𝑈12U_{1,2} by elements of Gε,nsubscript𝐺𝜀𝑛G_{\varepsilon,n} in order to obtain a subset of Gε,nsubscript𝐺𝜀𝑛G_{\varepsilon,n} that contains a positive density of elements of Bn(2)subscript𝐵𝑛superscript2B_{n}(\mathbb{Z}^{2}). Then, using the fact that 2superscript2\mathbb{Z}^{2} is abelian, we promote such a subset to all of Bn(2)subscript𝐵𝑛superscript2B_{n}(\mathbb{Z}^{2}) by taking sufficiently large sumsets in Proposition 4.9.

Lemma 4.7.

There exists δ>0superscript𝛿0\delta^{\prime}>0 with the following properties: for any ε>0𝜀0\varepsilon>0 there is an Nε>0superscriptsubscript𝑁𝜀0N_{\varepsilon}^{\prime}>0 such that for any nNε𝑛superscriptsubscript𝑁𝜀n\geq N_{\varepsilon}^{\prime} we have

|GUε,n|δ|Bn(2)|.𝐺subscript𝑈𝜀𝑛superscript𝛿subscript𝐵𝑛superscript2|GU_{\varepsilon,n}|\geq\delta^{\prime}|B_{n}(\mathbb{Z}^{2})|.
Proof.

Recall that Tksubscript𝑇𝑘T_{k} denotes the intersection of the ball of radius k𝑘k in SL(2,)H1,2similar-to-or-equalsSL2subscript𝐻12\mathrm{SL}(2,\mathbb{R})\simeq H_{1,2} with SL(2,)=Λ1,2SL2subscriptΛ12\mathrm{SL}(2,\mathbb{Z})=\Lambda_{1,2} and |Tk|subscript𝑇𝑘|T_{k}| denotes the cardinality of Tksubscript𝑇𝑘T_{k}. As |Tk|subscript𝑇𝑘|T_{k}| grows exponentially in k𝑘k, we may take s𝑠s fixed so that |Tks|<12|Tk|subscript𝑇𝑘𝑠12subscript𝑇𝑘|T_{k-s}|<\frac{1}{2}|T_{k}| for all k𝑘k sufficiently large. Given ε>0superscript𝜀0\varepsilon^{\prime}>0, define the subset SkSL(2,)subscript𝑆𝑘SL2S_{k}\subset\mathrm{SL}(2,\mathbb{Z}) to be

Sk:=Gε,kGε,k1(TkTks).assignsubscript𝑆𝑘subscript𝐺superscript𝜀𝑘superscriptsubscript𝐺superscript𝜀𝑘1subscript𝑇𝑘subscript𝑇𝑘𝑠S_{k}:=G_{\varepsilon^{\prime},k}\cap G_{\varepsilon^{\prime},k}^{-1}\cap(T_{k}\setminus T_{k-s}).

From Proposition 4.2, we may assume that

|Sk|12|Tk|.subscript𝑆𝑘12subscript𝑇𝑘|S_{k}|\geq\frac{1}{2}|T_{k}|.

From (3), there exists C1>0subscript𝐶10C_{1}>0 such that if A=[acbd]𝐴matrix𝑎𝑐𝑏𝑑A=\begin{bmatrix}a&c\\ b&d\end{bmatrix} belongs to Sksubscript𝑆𝑘S_{k} then either

(a,b)C1e12(ks)subscriptnorm𝑎𝑏subscript𝐶1superscript𝑒12𝑘𝑠\|(a,b)\|_{\infty}\geq C_{1}e^{\frac{1}{2}{(k-s)}} or (c,d)C1e12(ks)subscriptnorm𝑐𝑑subscript𝐶1superscript𝑒12𝑘𝑠\|(c,d)\|_{\infty}\geq C_{1}e^{\frac{1}{2}{(k-s)}}.

Without loss of generality, we assume that at least half of the elements in Sksubscript𝑆𝑘S_{k} satisfy (a,b)C1e12(ks)subscriptnorm𝑎𝑏subscript𝐶1superscript𝑒12𝑘𝑠\|(a,b)\|_{\infty}\geq C_{1}e^{\frac{1}{2}{(k-s)}}.

Consider the map P:Sk2:𝑃subscript𝑆𝑘superscript2P\colon S_{k}\to\mathbb{Z}^{2} that assigns A=[acbd]𝐴matrix𝑎𝑐𝑏𝑑A=\begin{bmatrix}a&c\\ b&d\end{bmatrix} to (a,b)𝑎𝑏(a,b). By (3), there is C2>1subscript𝐶21C_{2}>1 such that the image P(Sk)𝑃subscript𝑆𝑘P(S_{k}) of Sksubscript𝑆𝑘S_{k} lies in the norm-ball BC2ek2(2)subscript𝐵subscript𝐶2superscript𝑒𝑘2superscript2B_{C_{2}e^{\frac{k}{2}}}(\mathbb{Z}^{2}) for all k𝑘k.

Let k(n)=2log(n)logC2𝑘𝑛2𝑛subscript𝐶2k(n)=2\log(n)-\log C_{2}. Then P(Sk(n))Bn(2)𝑃subscript𝑆𝑘𝑛subscript𝐵𝑛superscript2P(S_{k(n)})\subset B_{n}(\mathbb{Z}^{2}). If n𝑛n is sufficiently large and A=[acbd]Sk(n)𝐴matrix𝑎𝑐𝑏𝑑subscript𝑆𝑘𝑛A=\begin{bmatrix}a&c\\ b&d\end{bmatrix}\in S_{k(n)} then we have ua,bGU(5ε,n)subscript𝑢𝑎𝑏𝐺subscript𝑈5superscript𝜀𝑛u_{a,b}\in GU_{(5\varepsilon^{\prime},n)}; indeed

α(ua,b)=α(A)α(u1,0)α(A1)𝛼subscript𝑢𝑎𝑏𝛼𝐴𝛼subscript𝑢10𝛼superscript𝐴1\alpha(u_{a,b})=\alpha(A)\circ\alpha(u_{1,0})\circ\alpha(A^{-1})

whence

Dα(ua,b)Dα(u1,0)e2εk(n).norm𝐷𝛼subscript𝑢𝑎𝑏norm𝐷𝛼subscript𝑢10superscript𝑒2superscript𝜀𝑘𝑛\|D\alpha(u_{a,b})\|\leq\|D\alpha(u_{1,0})\|e^{2\varepsilon^{\prime}k(n)}.

We have |Bn(2)|D1n2subscript𝐵𝑛superscript2subscript𝐷1superscript𝑛2|B_{n}(\mathbb{Z}^{2})|\leq D_{1}n^{2} for some D11subscript𝐷11D_{1}\geq 1. Also, from (4) and Lemma 4.3 we have |Sk(n)|12|Tk(n)|12ek(n)=1D2n2subscript𝑆𝑘𝑛12subscript𝑇𝑘𝑛12superscript𝑒𝑘𝑛1subscript𝐷2superscript𝑛2|S_{k(n)}|\geq\frac{1}{2}|T_{k(n)}|\geq\frac{1}{2}e^{k(n)}=\frac{1}{D_{2}}n^{2} for some D21subscript𝐷21D_{2}\geq 1.

To to complete the proof, we show that the preimage P1((a,b))superscript𝑃1𝑎𝑏P^{-1}((a,b)) in Sksubscript𝑆𝑘S_{k} of any (a,b)2𝑎𝑏superscript2(a,b)\in\mathbb{Z}^{2} satisfying (a,b)C1e12(ks)subscriptnorm𝑎𝑏subscript𝐶1superscript𝑒12𝑘𝑠\|(a,b)\|_{\infty}\geq C_{1}e^{\frac{1}{2}{(k-s)}} has uniformly bounded cardinality depending only on s𝑠s. Observe that if A,ASL(2,)𝐴superscript𝐴SL2A,A^{\prime}\in\mathrm{SL}(2,\mathbb{Z}) satisfy P(A)=P(A)𝑃𝐴𝑃superscript𝐴P(A)=P(A^{\prime}), then A=AUsuperscript𝐴𝐴𝑈A^{\prime}=AU, where U=[1m01]𝑈matrix1𝑚01U=\begin{bmatrix}1&m\\ 0&1\end{bmatrix} for some m𝑚m\in\mathbb{Z} and we have

A=[acbd],andA=[aam+cbbm+d].formulae-sequence𝐴matrix𝑎𝑐𝑏𝑑andsuperscript𝐴matrix𝑎𝑎𝑚𝑐𝑏𝑏𝑚𝑑A=\begin{bmatrix}a&c\\ b&d\end{bmatrix},\quad\text{and}\quad A^{\prime}=\begin{bmatrix}a&am+c\\ b&bm+d\end{bmatrix}.

If Asuperscript𝐴A^{\prime} belongs to Tksubscript𝑇𝑘T_{k} then (am+c,bm+d)C2ek2subscriptnorm𝑎𝑚𝑐𝑏𝑚𝑑subscript𝐶2superscript𝑒𝑘2\|(am+c,bm+d)\|_{\infty}\leq C_{2}e^{\frac{k}{2}} and if A𝐴A belongs to Tksubscript𝑇𝑘T_{k} then (c,d)C2ek2subscriptnorm𝑐𝑑subscript𝐶2superscript𝑒𝑘2\|(c,d)\|_{\infty}\leq C_{2}e^{\frac{k}{2}}. We thus have that |am|2C2ek2𝑎𝑚2subscript𝐶2superscript𝑒𝑘2|am|\leq 2C_{2}e^{{\frac{k}{2}}} and |bm|2C2ek2𝑏𝑚2subscript𝐶2superscript𝑒𝑘2|bm|\leq 2C_{2}e^{{\frac{k}{2}}}. As we assume that

(a,b)C1eks2subscriptnorm𝑎𝑏subscript𝐶1superscript𝑒𝑘𝑠2\|(a,b)\|_{\infty}\geq C_{1}e^{\frac{k-s}{2}}

we have that |m|2C2C1es2𝑚2subscript𝐶2subscript𝐶1superscript𝑒𝑠2|m|\leq 2\frac{C_{2}}{C_{1}}e^{{{\frac{s}{2}}}}. Thus, the preimage P1((a,b))superscript𝑃1𝑎𝑏P^{-1}((a,b)) has at most 4C2C1es2+14subscript𝐶2subscript𝐶1superscript𝑒𝑠214\frac{C_{2}}{C_{1}}e^{{{\frac{s}{2}}}}+1 elements in Sksubscript𝑆𝑘S_{k} .

With ε=15εsuperscript𝜀15𝜀\varepsilon^{\prime}=\frac{1}{5}\varepsilon, having taken n𝑛n sufficiently large, we thus have

|GUε,n||Bn(2)|1D1n212|Sk(n)|4C2C1es/2+1121D2n24C2C1es/2+11D1n2=:δ\displaystyle\frac{|GU_{\varepsilon,n}|}{|B_{n}(\mathbb{Z}^{2})|}\geq\frac{1}{D_{1}n^{2}}{\frac{\frac{1}{2}|S_{k(n)}|}{4\frac{C_{2}}{C_{1}}e^{s/2}+1}}\geq\frac{1}{2}\frac{\frac{1}{D_{2}}n^{2}}{4\frac{C_{2}}{C_{1}}e^{s/2}+1}\frac{1}{D_{1}n^{2}}=:\delta^{\prime}

which completes the proof.∎

To complete the proof of Proposition 4.6, we show that any element in Bn(2)subscript𝐵𝑛superscript2B_{n}(\mathbb{Z}^{2}) can be written as a product of a bounded number of elements in GUε,n𝐺subscript𝑈𝜀𝑛GU_{\varepsilon,n} independent of ε𝜀\varepsilon. This follows from the structure of sumsets of abelian groups.

From the chain rule and submultiplicativity of norms, we have the following.

Claim 4.8.

For any positive integers n,m𝑛𝑚n,m and ε1,ε2>0subscript𝜀1subscript𝜀20\varepsilon_{1},\varepsilon_{2}>0, if ua,bGUε1,nsubscript𝑢𝑎𝑏𝐺subscript𝑈subscript𝜀1𝑛u_{a,b}\in GU_{\varepsilon_{1},n} and uc,dGUε2,msubscript𝑢𝑐𝑑𝐺subscript𝑈subscript𝜀2𝑚u_{c,d}\in GU_{\varepsilon_{2},m} then the product ua,buc,dGUmax{ε1,ε2},n+msubscript𝑢𝑎𝑏subscript𝑢𝑐𝑑𝐺subscript𝑈subscript𝜀1subscript𝜀2𝑛𝑚u_{a,b}u_{c,d}\in GU_{\max\{\varepsilon_{1},\varepsilon_{2}\},n+m}

For subsets A,B2𝐴𝐵superscript2A,B\subset\mathbb{Z}^{2} we denote by A+B𝐴𝐵A+B the sumset of A,B𝐴𝐵A,B.

Claim 4.9.

For any 0<δ<10𝛿10<\delta<1, there exists a positive integer kδsubscript𝑘𝛿k_{\delta} and a finite set Fδ2subscript𝐹𝛿superscript2F_{\delta}\subset\mathbb{Z}^{2} such that for any n𝑛n and any symmetric set SnBn(2)subscript𝑆𝑛subscript𝐵𝑛superscript2S_{n}\subset B_{n}(\mathbb{Z}^{2}) with |Sn|>δ|Bn|subscript𝑆𝑛𝛿subscript𝐵𝑛|S_{n}|>\delta|B_{n}|, we have that

BnFδ+Sn+Sn++Snkδ times.subscript𝐵𝑛subscript𝐹𝛿subscriptsubscript𝑆𝑛subscript𝑆𝑛subscript𝑆𝑛kδ timesB_{n}\subset F_{\delta}+\underbrace{S_{n}+S_{n}+...+S_{n}}_{\text{$k_{\delta}$ times}}.
Proof.

Fix M+𝑀subscriptM\in\mathbb{Z}_{+} with 1M<δ1𝑀𝛿\frac{1}{M}<\delta. Take Nδ:=(M+1)!assignsubscript𝑁𝛿𝑀1N_{\delta}:=\big{(}M+1\big{)}!, kδ=4Nδsubscript𝑘𝛿4subscript𝑁𝛿k_{\delta}=4N_{\delta}, and Fδ:=BNδ(2).assignsubscript𝐹𝛿subscript𝐵subscript𝑁𝛿superscript2F_{\delta}:=B_{N_{\delta}}(\mathbb{Z}^{2}). Consider a symmetric set SnBn(2)subscript𝑆𝑛subscript𝐵𝑛superscript2S_{n}\subset B_{n}(\mathbb{Z}^{2}) with |Sn|>δ|Bn(2)|subscript𝑆𝑛𝛿subscript𝐵𝑛superscript2|S_{n}|>\delta|B_{n}(\mathbb{Z}^{2})|.

If nNδ𝑛subscript𝑁𝛿n\leq N_{\delta} then Bn(2)Fδsubscript𝐵𝑛superscript2subscript𝐹𝛿B_{n}(\mathbb{Z}^{2})\subset F_{\delta} and we are done. Thus, consider nNδ𝑛subscript𝑁𝛿n\geq N_{\delta}. To complete the proof the claim, we argue that the set

kδSn:=Sn+Sn++Snkδ timesassignsubscriptsubscript𝑘𝛿subscript𝑆𝑛subscriptsubscript𝑆𝑛subscript𝑆𝑛subscript𝑆𝑛kδ times\sum_{k_{\delta}}S_{n}:=\underbrace{S_{n}+S_{n}+...+S_{n}}_{\text{$k_{\delta}$ times}}

contains the intersection of the sublattice Nδ2subscript𝑁𝛿superscript2N_{\delta}\mathbb{Z}^{2} with Bn(2)subscript𝐵𝑛superscript2B_{n}(\mathbb{Z}^{2}). Adding Fδsubscript𝐹𝛿F_{\delta} to the sumset then implies the claim. Consider any non-zero vector v~Nδ2Bn(2)~𝑣subscript𝑁𝛿superscript2subscript𝐵𝑛superscript2\tilde{v}\in N_{\delta}\mathbb{Z}^{2}\cap B_{n}(\mathbb{Z}^{2}) of the form (~,0)~0(\tilde{\ell},0) for some ~[n,n]Nδ~𝑛𝑛subscript𝑁𝛿\tilde{\ell}\in[-n,n]\cap N_{\delta}\mathbb{Z}. Then v~=Nδv~𝑣subscript𝑁𝛿𝑣\tilde{v}=N_{\delta}v where v=(,0)𝑣0v=(\ell,0) is such that 0<||nNδ10𝑛superscriptsubscript𝑁𝛿10<|\ell|\leq\lfloor nN_{\delta}^{-1}\rfloor.

Consider the equivalence relation in Bn(2)subscript𝐵𝑛superscript2B_{n}(\mathbb{Z}^{2}) defined by declaring that two elements x,yR(n)𝑥𝑦𝑅𝑛x,y\in R(n) are equivalent if xy𝑥𝑦x-y is an integer multiple of v𝑣v. Each equivalence class is of the form

Cx={,xv,x,x+v,x+2v,.}.C_{x}=\{...,x-v,x,x+v,x+2v,....\}.

As |Sn|1M|Bn(2)|subscript𝑆𝑛1𝑀subscript𝐵𝑛superscript2|S_{n}|\geq\frac{1}{M}|B_{n}(\mathbb{Z}^{2})|, there exists one equivalence class Cxsubscript𝐶𝑥C_{x} such that |CxSn|1M|Cx|subscript𝐶𝑥subscript𝑆𝑛1𝑀subscript𝐶𝑥|C_{x}\cap S_{n}|\geq\frac{1}{M}|C_{x}|. Since 0<||nNδ10𝑛superscriptsubscript𝑁𝛿10<|\ell|\leq\lfloor nN_{\delta}^{-1}\rfloor, each equivalence class contains at least M+1𝑀1M+1 elements and hence CxSnsubscript𝐶𝑥subscript𝑆𝑛C_{x}\cap S_{n} contains at least two elements a,b𝑎𝑏a,b with b=a+iv𝑏𝑎𝑖𝑣b=a+iv for |i|M𝑖𝑀|i|\leq M. In particular, since ab=iv𝑎𝑏𝑖𝑣a-b=iv, we have ivSn+Sn𝑖𝑣subscript𝑆𝑛subscript𝑆𝑛iv\in S_{n}+S_{n}. As i𝑖i divides Nδsubscript𝑁𝛿N_{\delta}, we have that v~=Nδv2NδSn~𝑣subscript𝑁𝛿𝑣subscript2subscript𝑁𝛿subscript𝑆𝑛\tilde{v}=N_{\delta}v\in\sum_{2N_{\delta}}S_{n}.

Similarly, for nNδ𝑛subscript𝑁𝛿n\geq N_{\delta} and any u~Nδ2Bn(2)~𝑢subscript𝑁𝛿superscript2subscript𝐵𝑛superscript2\tilde{u}\in N_{\delta}\mathbb{Z}^{2}\cap B_{n}(\mathbb{Z}^{2}) of the form (0,~)0~(0,\tilde{\ell}) we have u~2NδSn~𝑢subscript2subscript𝑁𝛿subscript𝑆𝑛\tilde{u}\in\sum_{2N_{\delta}}S_{n}. Then

u~+v~4NδSn~𝑢~𝑣subscript4subscript𝑁𝛿subscript𝑆𝑛\tilde{u}+\tilde{v}\in\sum_{4N_{\delta}}S_{n}

completing the proof. ∎

Proof of Proposition 4.6.

Given ε>0superscript𝜀0\varepsilon^{\prime}>0, let δsuperscript𝛿\delta^{\prime} and Nεsuperscriptsubscript𝑁superscript𝜀N_{\varepsilon^{\prime}}^{\prime} be given by Lemma 4.7. Let Sn:=GUε,nassignsubscript𝑆𝑛𝐺subscript𝑈superscript𝜀𝑛S_{n}:=GU_{\varepsilon^{\prime},n} be as in (24) and take kδsuperscriptsubscript𝑘𝛿k_{\delta}^{\prime} and Fδsubscript𝐹superscript𝛿F_{\delta^{\prime}} as in Lemma 4.9. Note that GUε,n𝐺subscript𝑈superscript𝜀𝑛GU_{\varepsilon^{\prime},n} is symmetric by definition. Take NNε𝑁subscriptsuperscript𝑁superscript𝜀N\geq N^{\prime}_{\varepsilon^{\prime}} such that FδGUε,nsubscript𝐹superscript𝛿𝐺subscript𝑈superscript𝜀𝑛F_{\delta^{\prime}}\in GU_{\varepsilon^{\prime},n} whenever nN𝑛𝑁n\geq N. For nN𝑛𝑁n\geq N and any ua,bBn(2)subscript𝑢𝑎𝑏subscript𝐵𝑛superscript2u_{a,b}\in B_{n}(\mathbb{Z}^{2}) we have that ua,bFδ+Sn+Sn++Snsubscript𝑢𝑎𝑏subscript𝐹𝛿subscript𝑆𝑛subscript𝑆𝑛subscript𝑆𝑛u_{a,b}\in F_{\delta}+S_{n}+S_{n}+...+S_{n} (kδsubscript𝑘superscript𝛿k_{\delta^{\prime}} times) by Proposition 4.9. Proposition 4.8 then implies that ua,bGUε,(kδ+1)nsubscript𝑢𝑎𝑏𝐺subscript𝑈superscript𝜀subscript𝑘superscript𝛿1𝑛u_{a,b}\in GU_{\varepsilon^{\prime},(k_{\delta^{\prime}}+1)n} so D(ua,b)±1eεlog((kδ+1)n)norm𝐷superscriptsubscript𝑢𝑎𝑏plus-or-minus1superscript𝑒superscript𝜀subscript𝑘superscript𝛿1𝑛\|D(u_{a,b})^{\pm 1}\|\leq e^{\varepsilon^{\prime}\log((k_{\delta^{\prime}}+1)n)}. With ε=ε/2superscript𝜀𝜀2\varepsilon^{\prime}=\varepsilon/2, take Nεmax{N,(kδ+1)}.subscript𝑁𝜀𝑁subscript𝑘superscript𝛿1N_{\varepsilon}\geq\max\{N,(k_{\delta^{\prime}}+1)\}. Then for all nNε𝑛subscript𝑁𝜀n\geq N_{\varepsilon} we have

εlog((kδ+1)n)εlog(n)superscript𝜀subscript𝑘superscript𝛿1𝑛𝜀𝑛\varepsilon^{\prime}\log((k_{\delta^{\prime}}+1)n)\leq\varepsilon\log(n)

whence

D(ua,b)±1eεlog(n)norm𝐷superscriptsubscript𝑢𝑎𝑏plus-or-minus1superscript𝑒𝜀𝑛\|D(u_{a,b})^{\pm 1}\|\leq e^{\varepsilon\log(n)}

and for ua,bBn(2)subscript𝑢𝑎𝑏subscript𝐵𝑛superscript2u_{a,b}\in B_{n}(\mathbb{Z}^{2}) with nNε𝑛subscript𝑁𝜀n\geq N_{\varepsilon}. ∎

5. Proof of Theorem B

5.1. Reduction to the restriction of an action by Λi,jsubscriptΛ𝑖𝑗\Lambda_{i,j}

We recall the work of Lubotzky, Mozes, and Raghunathan, namely [LMR1] and [LMR2], which establishes quasi-isometry between the word and Riemannian metrics on lattices in higher-rank semisimple Lie groups. In the special case of Γ=SL(m,)ΓSL𝑚\Gamma=\mathrm{SL}(m,\mathbb{Z}) for m3𝑚3m\geq 3, in [LMR1, Corollary 3] it is shown that any element γ𝛾\gamma of SL(m,)SL𝑚\mathrm{SL}(m,\mathbb{Z}) is written as a product of at most m2superscript𝑚2m^{2} elements γisubscript𝛾𝑖\gamma_{i}. Moreover each γisubscript𝛾𝑖\gamma_{i} is contained some Λi,jSL(2,)similar-to-or-equalssubscriptΛ𝑖𝑗SL2\Lambda_{i,j}\simeq\mathrm{SL}(2,\mathbb{Z}) and the word-length of each γisubscript𝛾𝑖\gamma_{i} is proportional to the word-length of γ𝛾\gamma.

Thus, to establish that an action α:ΓDiff1(M):𝛼ΓsuperscriptDiff1𝑀\alpha\colon\Gamma\to\operatorname{Diff}^{1}(M) has uniform subexponential growth of derivatives in Theorem B, it is sufficient to show that the restriction α|Λi,j:ΓDiff1(M):evaluated-at𝛼subscriptΛ𝑖𝑗ΓsuperscriptDiff1𝑀{\alpha}{|_{{\Lambda_{i,j}}}}\colon\Gamma\to\operatorname{Diff}^{1}(M) has uniform subexponential growth of derivatives for each 1ijm1𝑖𝑗𝑚1\leq i\neq j\leq m. We emphasize that to measure subexponential growth of derivatives, the word-length on Λi,jsubscriptΛ𝑖𝑗{\Lambda_{i,j}} is measured as the word-length as embedded in SL(m,)SL𝑚\mathrm{SL}(m,\mathbb{Z}) (which is quasi-isometric to the Riemannian metric on SL(m,)SL𝑚\mathrm{SL}(m,\mathbb{R})) rather than the intrinsic word-length in Λi,jSL(2,)similar-to-or-equalssubscriptΛ𝑖𝑗SL2\Lambda_{i,j}\simeq\mathrm{SL}(2,\mathbb{Z}) (which is not quasi-isometric to the Riemannian metric on SL(2,)SL2\mathrm{SL}(2,\mathbb{R})).

As the Weyl group acts transitively on the set of all Λi,jsubscriptΛ𝑖𝑗\Lambda_{i,j}, it is sufficient to consider a fixed Λi,jsubscriptΛ𝑖𝑗\Lambda_{i,j}. Thus to deduce Theorem B, in the remainder of this section we establish the following, which is the main proposition of the paper.

Proposition 5.1.

For any action α:ΓDiff1(M):𝛼ΓsuperscriptDiff1𝑀\alpha\colon\Gamma\to\operatorname{Diff}^{1}(M) as in Theorem B, the restricted action α|Λ1,2:ΓDiff1(M):evaluated-at𝛼subscriptΛ12ΓsuperscriptDiff1𝑀{\alpha}{|_{{\Lambda_{1,2}}}}\colon\Gamma\to\operatorname{Diff}^{1}(M) has uniform subexponential growth of derivatives.

5.2. Orbits with large fiber growth yet low depth in the cusp

To prove Proposition 5.1, as in Section 4.2 we consider a canonical embedding X=H1,2/Λ1,2𝑋subscript𝐻12subscriptΛ12X=H_{1,2}/\Lambda_{1,2} of SL(2,)/SL(2,)SL2SL2\mathrm{SL}(2,\mathbb{R})/\mathrm{SL}(2,\mathbb{Z}) in SL(m,)/SL(m,)SL𝑚SL𝑚\mathrm{SL}(m,\mathbb{R})/\mathrm{SL}(m,\mathbb{Z}). Write

at:=diag(et/2,et/2)SL(2,)assignsuperscript𝑎𝑡diagsuperscript𝑒𝑡2superscript𝑒𝑡2SL2a^{t}:=\text{diag}(e^{t/2},e^{-t/2})\subset\mathrm{SL}(2,\mathbb{R})

for the geodesic flow on X𝑋X. Let Xthicksubscript𝑋thickX_{\text{thick}} be a fixed compact SO(2)SO2\mathrm{SO}(2)-invariant “thick part” of X𝑋X; that is, relative to the Dirichlet domain 𝒟𝒟\mathcal{D} in (7), points in SO(2)\Xthick\SO2subscript𝑋thick\mathrm{SO}(2)\backslash X_{\text{thick}} corresponds to the points in SO(2)\𝒟\SO2𝒟\mathrm{SO}(2)\backslash\mathcal{D} whose imaginary part is bounded above, say, by 171717.

A geodesic curve in the modular surface of length t𝑡t corresponds to the image of an orbit ζ={as(x)}0st𝜁subscriptsuperscript𝑎𝑠𝑥0𝑠𝑡\zeta=\{a^{s}(x)\}_{0\leq s\leq t} where xX𝑥𝑋x\in X and t0𝑡0t\geq 0. Denote the length of such a curve by l(ζ)𝑙𝜁l(\zeta). For an orbit ζ={as(x)}0st𝜁subscriptsuperscript𝑎𝑠𝑥0𝑠𝑡\zeta=\{a^{s}(x)\}_{0\leq s\leq t} of {at}superscript𝑎𝑡\{a^{t}\} in X𝑋X we define

c(ζ):=log(Dx(at)Fiber).assign𝑐𝜁subscriptnormsubscript𝐷𝑥superscript𝑎𝑡Fiberc(\zeta):=\log(\|D_{x}(a^{t})\|_{\text{Fiber}}).

The following claim is straightforward from the compactness Xthicksubscript𝑋thickX_{\text{thick}} and the quasi-isometry between the word and Riemannian metrics on ΓΓ\Gamma.

Claim 5.2.

For an action α:SL(m,)Diff1(M):𝛼SL𝑚superscriptDiff1𝑀\alpha\colon\mathrm{SL}(m,\mathbb{Z})\to\mathrm{Diff}^{1}(M), the following statements are equivalent:

  1. (1)

    the restriction α|Λ1,2:Λ1,2Diff1(M):evaluated-at𝛼subscriptΛ12subscriptΛ12superscriptDiff1𝑀{\alpha}{|_{{\Lambda_{1,2}}}}\colon\Lambda_{1,2}\to\mathrm{Diff}^{1}(M) has uniform subexponential growth of derivatives;

  2. (2)

    for any ε>0𝜀0\varepsilon>0 there is a tε>0subscript𝑡𝜀0t_{\varepsilon}>0 such that for any orbit ζ={as(x)}0st𝜁subscriptsuperscript𝑎𝑠𝑥0𝑠𝑡\zeta=\{a^{s}(x)\}_{0\leq s\leq t} with xXthick𝑥subscript𝑋thickx\in X_{\mathrm{thick}}, at(x)Xthicksuperscript𝑎𝑡𝑥subscript𝑋thicka^{t}(x)\in X_{\mathrm{thick}}, and l(ζ)=ttε𝑙𝜁𝑡subscript𝑡𝜀l(\zeta)=t\geq{t_{\varepsilon}} we have

    c(ζ)εl(ζ).𝑐𝜁𝜀𝑙𝜁c(\zeta)\leq\varepsilon l(\zeta).

Define the maximal fiberwise growth rate of orbits starting and returning to Xthicksubscript𝑋thickX_{\mathrm{thick}} to be

χmax:=lim supt>0{sup{logDx(at)|Fibert:xXthick,at(x)Xthick}}.\chi_{\mathrm{max}}:=\limsup_{t>0}\left\{\sup\left\{\frac{\log\|{D_{x}(a^{t})}{|_{{{\text{Fiber}}}}}\|}{t}:x\in X_{\mathrm{thick}},a^{t}(x)\in X_{\mathrm{thick}}\right\}\right\}. (25)

Using Claim 5.2, to establish Proposition 5.1 it is sufficient to show that χmax=0subscript𝜒max0\chi_{\mathrm{max}}=0.

For an orbit ζ={as(x)}0st𝜁subscriptsuperscript𝑎𝑠𝑥0𝑠𝑡\zeta=\{a^{s}(x)\}_{0\leq s\leq t}, define the following function which measures the depth of ζ𝜁\zeta into the cusp:

d(ζ)=max0stdist(as(x),Xthick).𝑑𝜁subscript0𝑠𝑡distsuperscript𝑎𝑠𝑥subscript𝑋thickd(\zeta)=\max_{0\leq s\leq t}\text{dist}(a^{s}(x),X_{\mathrm{thick}}).

The following lemma is the main result of this subsection.

Lemma 5.3.

If χmax>0subscript𝜒max0\chi_{\text{max}}>0 then there exists a sequence of orbits ζn={as(xn)}0stnsubscript𝜁𝑛subscriptsuperscript𝑎𝑠subscript𝑥𝑛0𝑠subscript𝑡𝑛\zeta_{n}=\{a^{s}(x_{n})\}_{0\leq s\leq t_{n}} with xnXthicksubscript𝑥𝑛subscript𝑋thickx_{n}\in X_{\mathrm{thick}}, atn(xn)Xthicksuperscript𝑎subscript𝑡𝑛subscript𝑥𝑛subscript𝑋thicka^{t_{n}}(x_{n})\in X_{\mathrm{thick}}, and tn=l(ζn)subscript𝑡𝑛𝑙subscript𝜁𝑛t_{n}=l(\zeta_{n})\to\infty such that

  1. (1)

    c(ζn)χmax2tn𝑐subscript𝜁𝑛subscript𝜒max2subscript𝑡𝑛\displaystyle c(\zeta_{n})\geq\frac{\chi_{\mathrm{max}}}{2}t_{n};

  2. (2)

    limnd(ζn)tn=0subscript𝑛𝑑subscript𝜁𝑛subscript𝑡𝑛0\displaystyle\lim_{n\to\infty}\frac{d(\zeta_{n})}{t_{n}}=0.

We first have the following claim.

Claim 5.4.

For any ε>0𝜀0\varepsilon>0 there exists tεsubscript𝑡𝜀t_{\varepsilon} with the following properties: for any xXthick𝑥subscript𝑋thickx\in\partial X_{\mathrm{thick}} and ttε𝑡subscript𝑡𝜀t\geq t_{\varepsilon} such that as(x)XXthicksuperscript𝑎𝑠𝑥𝑋subscript𝑋thicka^{s}(x)\in X\smallsetminus X_{\mathrm{thick}} for all 0<s<t0𝑠𝑡0<s<t and at(x)Xthicksuperscript𝑎𝑡𝑥subscript𝑋thicka^{t}(x)\in\partial X_{\mathrm{thick}} then, for the orbit ζ={as(x)}0st𝜁subscriptsuperscript𝑎𝑠𝑥0𝑠𝑡\zeta=\{a^{s}(x)\}_{0\leq s\leq t}, we have

c(ζ)εt=εl(ζ).𝑐𝜁𝜀𝑡𝜀𝑙𝜁c(\zeta)\leq\varepsilon t=\varepsilon l(\zeta).

Indeed, the claim follows from the fact that the value of the return cocycle β(as,x)𝛽superscript𝑎𝑠𝑥\beta(a^{s},x) is defined by geodesic in the cusp of X𝑋X is given by a unipotent matrix of the form (1n01)Λ1,2SL(m,)1𝑛01subscriptΛ12SL𝑚\left(\begin{array}[]{cc}1&n\\ 0&1\end{array}\right)\in\Lambda_{1,2}\subset\mathrm{SL}(m,\mathbb{Z}) and Proposition 4.1.

Proof of Lemma 5.3.

Let ζn:={as(xn)}0stnassignsubscript𝜁𝑛subscriptsuperscript𝑎𝑠subscript𝑥𝑛0𝑠subscript𝑡𝑛\zeta_{n}:=\{a^{s}(x_{n})\}_{0\leq s\leq t_{n}} be a sequence of orbits with xnXthicksubscript𝑥𝑛subscript𝑋thickx_{n}\in X_{\mathrm{thick}}, atn(xn)Xthicksuperscript𝑎subscript𝑡𝑛subscript𝑥𝑛subscript𝑋thicka^{t_{n}}(x_{n})\in X_{\mathrm{thick}}, tnsubscript𝑡𝑛t_{n}\to\infty, and such that

χmax=limnc(ζn)tn.subscript𝜒subscript𝑛𝑐subscript𝜁𝑛subscript𝑡𝑛\chi_{\max}=\lim_{n\to\infty}\frac{c(\zeta_{n})}{t_{n}}.

Replacing ζnsubscript𝜁𝑛\zeta_{n} with a subsequence, we may assume the following limit exists:

β:=limnd(ζn)tn.assign𝛽subscript𝑛𝑑subscript𝜁𝑛subscript𝑡𝑛\beta:=\lim_{n\to\infty}\frac{d(\zeta_{n})}{t_{n}}.

We aim to prove that β=0𝛽0\beta=0. Arguing by contradiction, suppose 0<β10𝛽10<\beta\leq 1. We decompose the orbit

ζn=αknωkn1αkn1ω1α1subscript𝜁𝑛subscript𝛼subscript𝑘𝑛subscript𝜔subscript𝑘𝑛1subscript𝛼subscript𝑘𝑛1subscript𝜔1subscript𝛼1\zeta_{n}=\alpha_{k_{n}}\omega_{k_{n-1}}\alpha_{k_{n-1}}\cdots\omega_{1}\alpha_{1}

as a concatenation of smaller orbit segments αi,ωisubscript𝛼𝑖subscript𝜔𝑖\alpha_{i},\omega_{i} with the following properties:

  1. (1)

    each orbit αisubscript𝛼𝑖\alpha_{i} is such that d(αi)β2tn𝑑subscript𝛼𝑖𝛽2subscript𝑡𝑛d(\alpha_{i})\leq\frac{\beta}{2}t_{n};

  2. (2)

    the endpoints of each orbit αisubscript𝛼𝑖\alpha_{i} are contained in Xthicksubscript𝑋thickX_{\mathrm{thick}};

  3. (3)

    each orbit ωisubscript𝜔𝑖\omega_{i} is contained entirely in (XXthick)Xthick𝑋subscript𝑋thicksubscript𝑋thick(X\smallsetminus X_{\mathrm{thick}})\cup\partial X_{\mathrm{thick}} with endpoints contained in Xthicksubscript𝑋thick\partial X_{\mathrm{thick}};

  4. (4)

    each orbit ωisubscript𝜔𝑖\omega_{i} satisfies d(ωi)β2tn𝑑subscript𝜔𝑖𝛽2subscript𝑡𝑛d(\omega_{i})\geq\frac{\beta}{2}t_{n} whence l(ωi)β2tn𝑙subscript𝜔𝑖𝛽2subscript𝑡𝑛l(\omega_{i})\geq\frac{\beta}{2}t_{n} for tnsubscript𝑡𝑛t_{n} sufficiently large.

Note for each n𝑛n, that kn2β+1subscript𝑘𝑛2𝛽1k_{n}\leq\lfloor\frac{2}{\beta}\rfloor+1 and thus knsubscript𝑘𝑛k_{n} is bounded by some k𝑘k independent of n𝑛n. Additionally, since SL(m,)SL𝑚\mathrm{SL}(m,\mathbb{Z}) is finitely generated and (equipped with the word metric) is quasi-isometrically embedded in SL(m,)SL𝑚\mathrm{SL}(m,\mathbb{R}), there exists a constant K𝐾K such that for any orbit segment ζ𝜁\zeta whose endpoints are contained in Xthicksubscript𝑋thickX_{\mathrm{thick}}, we have c(ζ)Kl(ζ)𝑐𝜁𝐾𝑙𝜁c(\zeta)\leq Kl(\zeta). By the definition of χmaxsubscript𝜒\chi_{\max}, for any ε>0𝜀0\varepsilon>0 there is a positive constant Mεsubscript𝑀𝜀M_{\varepsilon} such that for any orbit sub-segment αisubscript𝛼𝑖\alpha_{i}

  1. (1)

    c(αi)(χmax+ε)l(αi)𝑐subscript𝛼𝑖subscript𝜒𝜀𝑙subscript𝛼𝑖c(\alpha_{i})\leq(\chi_{\max}+\varepsilon)l(\alpha_{i}) whenever l(αi)>Mε𝑙subscript𝛼𝑖subscript𝑀𝜀l(\alpha_{i})>M_{\varepsilon}

  2. (2)

    c(αi)KMε𝑐subscript𝛼𝑖𝐾subscript𝑀𝜀c(\alpha_{i})\leq KM_{\varepsilon} whenever l(αi)Mε𝑙subscript𝛼𝑖subscript𝑀𝜀l(\alpha_{i})\leq M_{\varepsilon}.

From Claim 5.4, for any ε>0𝜀0\varepsilon>0 we have, assuming that n𝑛n and hence tnsubscript𝑡𝑛t_{n} are sufficiently large, that

c(ωi)<εl(ωi)𝑐subscript𝜔𝑖𝜀𝑙subscript𝜔𝑖c(\omega_{i})<\varepsilon l(\omega_{i})

for all orbit sub-segmants ωisubscript𝜔𝑖\omega_{i}.

Taking n𝑛n sufficiently large we have

(χmaxε)tn<c(ζn)ic(ωi)+ic(αi).subscript𝜒𝜀subscript𝑡𝑛𝑐subscript𝜁𝑛subscript𝑖𝑐subscript𝜔𝑖subscript𝑖𝑐subscript𝛼𝑖(\chi_{\max}-\varepsilon)t_{n}<c(\zeta_{n})\leq\sum_{i}c(\omega_{i})+\sum_{i}c(\alpha_{i}). (26)

As we assume β>0𝛽0\beta>0, for all sufficiently large n𝑛n there exists at least one orbit sub-segment ωisubscript𝜔𝑖\omega_{i} and thus for such n𝑛n

ic(αi)kKMε+(χmax+ε)il(αi)kKMε+(χmax+ε)(1β/2)tn.subscript𝑖𝑐subscript𝛼𝑖𝑘𝐾subscript𝑀𝜀subscript𝜒𝜀subscript𝑖𝑙subscript𝛼𝑖𝑘𝐾subscript𝑀𝜀subscript𝜒𝜀1𝛽2subscript𝑡𝑛\sum_{i}c(\alpha_{i})\leq kKM_{\varepsilon}+(\chi_{\max}+\varepsilon)\sum_{i}l(\alpha_{i})\leq kKM_{\varepsilon}+(\chi_{\max}+\varepsilon)(1-\beta/2)t_{n}. (27)

From (26) and (27) we obtain that

(χmaxε)tn(εtn)+(kKMε+(χmax+ε)(1β/2)tn).subscript𝜒𝜀subscript𝑡𝑛𝜀subscript𝑡𝑛𝑘𝐾subscript𝑀𝜀subscript𝜒𝜀1𝛽2subscript𝑡𝑛(\chi_{\max}-\varepsilon)t_{n}\leq(\varepsilon t_{n})+\Big{(}kKM_{\varepsilon}+(\chi_{\max}+\varepsilon)(1-\beta/2)t_{n}\Big{)}. (28)

Dividing by tnsubscript𝑡𝑛t_{n} and taking n𝑛n\to\infty obtain

χmaxεε+(χmax+ε)(1β/2).subscript𝜒𝜀𝜀subscript𝜒𝜀1𝛽2\chi_{\max}-\varepsilon\leq\varepsilon+(\chi_{\max}+\varepsilon)(1-\beta/2).

As we assumed χmax>0subscript𝜒0\chi_{\max}>0 and β>0𝛽0\beta>0, we obtain a contradiction by taking ε>0𝜀0\varepsilon>0 sufficiently small. ∎

5.3. Construction of a Følner sequence and family averaged measures

Assuming that χmaxsubscript𝜒max\chi_{\mathrm{max}} in (25) is non-zero, we start from the orbit segments constructed in Lemma 5.3 and perform an averaging procedure to obtain a family of measures {μn}subscript𝜇𝑛\{\mu_{n}\} on Mαsuperscript𝑀𝛼M^{\alpha} whose properties lead to a contradiction. In particular, the projection of any weak-* limit μsubscript𝜇\mu_{\infty} of μnsubscript𝜇𝑛\mu_{n} to Mαsuperscript𝑀𝛼M^{\alpha} will be A𝐴A-invariant, well behaved at the cusps, and have non-zero Lyapunov exponents. These measures on Mαsuperscript𝑀𝛼M^{\alpha} are obtained by averaging certain Dirac measures against Følner sequences in a certain amenable subgroup of G𝐺G.

Consider the copy of SL(m1,)SL(m,)SL𝑚1SL𝑚\mathrm{SL}(m-1,\mathbb{R})\subset\mathrm{SL}(m,\mathbb{R}) as the subgroup of matrices that differ from the identity away from the m𝑚mth row and m𝑚mth column. Let Nm1similar-to-or-equalssuperscript𝑁superscript𝑚1N^{\prime}\simeq\mathbb{R}^{m-1} be the abelian subgroup of unipotent elements that differ from the identity only in the m𝑚mth column; that is given a vector r=(r1,r2,,rm1)m1𝑟subscript𝑟1subscript𝑟2subscript𝑟𝑚1superscript𝑚1r=(r_{1},r_{2},\dots,r_{m-1})\in\mathbb{R}^{m-1} define ursuperscript𝑢𝑟u^{r} to be the unipotent element

ur=(100r110r21rm11)superscript𝑢𝑟100subscript𝑟1missing-subexpression10subscript𝑟2missing-subexpressionmissing-subexpressionmissing-subexpressionmissing-subexpressionmissing-subexpressionmissing-subexpression1subscript𝑟𝑚1missing-subexpressionmissing-subexpressionmissing-subexpressionmissing-subexpression1u^{r}=\left(\begin{array}[]{ccccc}1&0&0&\dots&r_{1}\\ &1&0&\dots&r_{2}\\ &&\ddots&&\vdots\\ &&&1&r_{m-1}\\ &&&&1\end{array}\right) (29)

and let N={ur}superscript𝑁superscript𝑢𝑟N^{\prime}=\{u^{r}\}. Nsuperscript𝑁N^{\prime} is normalized by SL(m1,)SL𝑚1\mathrm{SL}(m-1,\mathbb{R}).

Identifying Nsuperscript𝑁N^{\prime} with m1superscript𝑚1\mathbb{R}^{m-1} we have an embedding SL(m1,)m1SL(m,)left-normal-factor-semidirect-productSL𝑚1superscript𝑚1SL𝑚\mathrm{SL}(m-1,\mathbb{R})\ltimes\mathbb{R}^{m-1}\subset\mathrm{SL}(m,\mathbb{R}). The subgroup SL(m1,)m1left-normal-factor-semidirect-productSL𝑚1superscript𝑚1\mathrm{SL}(m-1,\mathbb{R})\ltimes\mathbb{R}^{m-1} has as a lattice the subgroup

SL(m1,)m1:=Γ(SL(m1,)m1)assignleft-normal-factor-semidirect-productSL𝑚1superscript𝑚1Γleft-normal-factor-semidirect-productSL𝑚1superscript𝑚1\mathrm{SL}(m-1,\mathbb{Z})\ltimes\mathbb{Z}^{m-1}:=\Gamma\cap\big{(}\mathrm{SL}(m-1,\mathbb{R})\ltimes\mathbb{R}^{m-1}\big{)}

and there is a natural embedding given by the inclusion

(SL(m1,)m1)/(SL(m1,1)m1))SL(m,)/SL(m,).(\mathrm{SL}(m-1,\mathbb{R})\ltimes\mathbb{R}^{m-1})/(\mathrm{SL}(m-1,\mathbb{Z}^{-1})\ltimes\mathbb{Z}^{m-1}))\subset\mathrm{SL}(m,\mathbb{R})/\mathrm{SL}(m,\mathbb{Z}).

Recall A𝐴A is the group of diagonal matrices with positive entries. Let at,bsAsuperscript𝑎𝑡superscript𝑏𝑠𝐴a^{t},b^{s}\in A denote matrices

at=diag(et/2,et/2,1,1,1)superscript𝑎𝑡diagsuperscript𝑒𝑡2superscript𝑒𝑡2111a^{t}=\text{diag}(e^{t/2},e^{-t/2},1,1...,1)
bs=diag(es,es,es..,es,es(m1)).b^{s}=\text{diag}(e^{s},e^{s},e^{s}..,e^{s},e^{-s(m-1)}).

Complete the set {a,b}𝑎𝑏\{a,b\} to a spanning set {a,b,c1,c2cm3}𝑎𝑏subscript𝑐1subscript𝑐2subscript𝑐𝑚3\{a,b,c_{1},c_{2}\dots c_{m-3}\} of A𝐴A viewed as vector space where the cisubscript𝑐𝑖c_{i} are diagonal matrices whose (m,m)𝑚𝑚(m,m)-entry is equal to 111.

Let FnANsubscript𝐹𝑛𝐴superscript𝑁F_{n}\subset AN^{\prime} be the subset of G𝐺G consisting of all the elements of the form

atbsc=1m3cisiursuperscript𝑎𝑡superscript𝑏𝑠superscriptsubscriptproduct𝑐1𝑚3superscriptsubscript𝑐𝑖subscript𝑠𝑖superscript𝑢𝑟a^{t}b^{s}\prod_{c=1}^{m-3}c_{i}^{s_{i}}u^{r} (30)

where, for some δ>0𝛿0\delta>0 to be determined later (in the proof of Proposition 5.10 below),

  1. (1)

    0<t<tn0𝑡subscript𝑡𝑛0<t<t_{n};

  2. (2)

    δtn/2<s<δtn𝛿subscript𝑡𝑛2𝑠𝛿subscript𝑡𝑛\delta t_{n}/2<s<\delta t_{n};

  3. (3)

    0<si<tn0subscript𝑠𝑖subscript𝑡𝑛0<s_{i}<\sqrt{t_{n}};

  4. (4)

    rBm1(e200tn)𝑟subscript𝐵superscript𝑚1superscript𝑒200subscript𝑡𝑛r\in B_{\mathbb{R}^{m-1}}(e^{200t_{n}}).

Claim 5.5.

{Fn}subscript𝐹𝑛\{F_{n}\} is Følner sequence in AN𝐴superscript𝑁AN^{\prime}.

Observe that Fnsubscript𝐹𝑛F_{n} is linearly-long in the a𝑎a-direction and exponentially-long in the Nsuperscript𝑁N^{\prime}-direction. From conditions (2) and (4), the A𝐴A-component of Fnsubscript𝐹𝑛F_{n} is much longer in the atsuperscript𝑎𝑡a^{t}-direction than in the other directions. The condition (2) that δtn/2<s𝛿subscript𝑡𝑛2𝑠\delta t_{n}/2<s is fundamental in our estimates in Section 5.4 that ensure the measures constructed below {μn}subscript𝜇𝑛\{\mu_{n}\} have uniformly exponentially small mass in the cusps. These estimates are related to the fact that orbits of Nsuperscript𝑁N^{\prime} correspond to the unstable manifolds for the flow defined by bssubscript𝑏𝑠b_{s} in SL(m,)/SL(m,)SL𝑚SL𝑚\mathrm{SL}(m,\mathbb{R})/\mathrm{SL}(m,\mathbb{Z}) and open subsets of unstable manifolds equidistribute to the Haar measure on SL(m,)/SL(m,)SL𝑚SL𝑚\mathrm{SL}(m,\mathbb{R})/\mathrm{SL}(m,\mathbb{Z}) under the flow bssuperscript𝑏𝑠b^{s}.

Recall we have a sequence of fiber bundles

FMαG/Γ𝐹superscript𝑀𝛼𝐺ΓF\to M^{\alpha}\to G/\Gamma

and may consider F𝐹F as a fiber bundle over G/Γ𝐺ΓG/\Gamma. Given xG/Γ𝑥𝐺Γx\in G/\Gamma, let F(x)TMsimilar-to-or-equals𝐹𝑥𝑇𝑀F(x)\simeq TM denote the fiber of F𝐹F over x𝑥x. An element vF(x)𝑣𝐹𝑥v\in F(x) is a pair v=(y,ξ)𝑣𝑦𝜉v=(y,\xi) where, identifying the fiber of Mαsuperscript𝑀𝛼M^{\alpha} through x𝑥x with M𝑀M, we have yM𝑦𝑀y\in M and ξTyM𝜉subscript𝑇𝑦𝑀\xi\in T_{y}M. Given v=(y,ξ)F(x)𝑣𝑦𝜉𝐹𝑥v=(y,\xi)\in F(x), we write v=ξnorm𝑣norm𝜉\|v\|=\|\xi\| using our chosen norm on F𝐹F. Given v=(y,ξ)F(x)𝑣𝑦𝜉𝐹𝑥v=(y,\xi)\in F(x), let p(v)=y𝑝𝑣𝑦p(v)=y denote the footpoint of v𝑣v in the fiber of Mαsuperscript𝑀𝛼M^{\alpha} through x𝑥x.

If uniform subexponential growth of derivatives fails for the restriction of the α𝛼\alpha to Λ1,2subscriptΛ12\Lambda_{1,2}, then there exist sequences xnXthicksubscript𝑥𝑛subscript𝑋thickx_{n}\in X_{\mathrm{thick}}, vnF(xn)subscript𝑣𝑛𝐹subscript𝑥𝑛v_{n}\in F(x_{n}) with vn=1normsubscript𝑣𝑛1\|v_{n}\|=1, and tnsubscript𝑡𝑛t_{n}\in\mathbb{R} as in Lemma 5.3 and Claim 5.2 with tnsubscript𝑡𝑛t_{n}\to\infty, such that

Dxnantn(vn)eλtnnormsubscript𝐷subscript𝑥𝑛superscriptsubscript𝑎𝑛subscript𝑡𝑛subscript𝑣𝑛superscript𝑒𝜆subscript𝑡𝑛\|D_{x_{n}}a_{n}^{t_{n}}(v_{n})\|\geq e^{\lambda t_{n}} (31)

for some λ>0𝜆0\lambda>0.

Note that AN𝐴superscript𝑁AN^{\prime} is a solvable group. We may equip AN𝐴superscript𝑁AN^{\prime} with any left-invariant Haar measure. Note that the ambient Riemannian metric induces a right-invariant Haar measure on AN𝐴superscript𝑁AN^{\prime} but as AN𝐴superscript𝑁AN^{\prime} is not unimodular these measures do not coincide.

For each n𝑛n, take μnsubscript𝜇𝑛\mu_{n} to be the measure on Mαsuperscript𝑀𝛼M^{\alpha} obtained by averaging the Dirac measure δ(xn,p(vn))subscript𝛿subscript𝑥𝑛𝑝subscript𝑣𝑛\delta_{(x_{n},p(v_{n}))} over the set Fnsubscript𝐹𝑛F_{n}:

μn:=1|Fn|Fngδ(xn,p(vn))𝑑gassignsubscript𝜇𝑛1subscriptsubscript𝐹𝑛subscriptsubscript𝐹𝑛𝑔𝛿subscript𝑥𝑛𝑝subscript𝑣𝑛differential-d𝑔\mu_{n}:=\frac{1}{|F_{n}|_{\ell}}\int_{F_{n}}g\cdot\delta(x_{n},p(v_{n}))\ dg

where |Fn|subscriptsubscript𝐹𝑛|F_{n}|_{\ell} is the volume of Fnsubscript𝐹𝑛F_{n} and dg𝑑𝑔dg indicates integration with respect to left-invariant Haar measure on AN𝐴superscript𝑁AN^{\prime}.

We expand the above integral in our coordinates introduced above. Then for any bounded continuous function f:Mα:𝑓superscript𝑀𝛼f\colon M^{\alpha}\to\mathbb{R}, integrating against our Euclidean parameters t,s,si,𝑡𝑠subscript𝑠𝑖t,s,s_{i}, and r𝑟r we have

Mαsubscriptsuperscript𝑀𝛼\displaystyle\int\limits_{M^{\alpha}} fdμn𝑓𝑑subscript𝜇𝑛\displaystyle f\ d\mu_{n}\quad\quad (32)
=20tnδtn/2δtn[0,tn]m3Bm1(e200tn)f(atbsc=1m3cisiur(xn,p(vn)))𝑑r𝑑si𝑑s𝑑ttnδtntnm3|Bm1(e200tn)|absent2superscriptsubscript0subscript𝑡𝑛superscriptsubscript𝛿subscript𝑡𝑛2𝛿subscript𝑡𝑛subscriptsuperscript0subscript𝑡𝑛𝑚3subscriptsubscript𝐵superscript𝑚1superscript𝑒200subscript𝑡𝑛𝑓superscript𝑎𝑡superscript𝑏𝑠superscriptsubscriptproduct𝑐1𝑚3superscriptsubscript𝑐𝑖subscript𝑠𝑖superscript𝑢𝑟subscript𝑥𝑛𝑝subscript𝑣𝑛differential-d𝑟differential-dsubscript𝑠𝑖differential-d𝑠differential-d𝑡subscript𝑡𝑛𝛿subscript𝑡𝑛superscriptsubscript𝑡𝑛𝑚3subscript𝐵superscript𝑚1superscript𝑒200subscript𝑡𝑛\displaystyle=\frac{\displaystyle 2\int\limits_{0}^{t_{n}}\int\limits_{\delta t_{n}/2}^{\delta t_{n}}\int\limits_{[0,\sqrt{t_{n}}]^{m-3}}\int\limits_{B_{\mathbb{R}^{m-1}}(e^{200t_{n}})}f\left(a^{t}b^{s}\prod_{c=1}^{m-3}c_{i}^{s_{i}}u^{r}\cdot(x_{n},p(v_{n}))\right)\ dr\ d{s_{i}}\ ds\ dt}{{t_{n}}{\delta t_{n}}{\sqrt{t_{n}}}^{m-3}{|B_{\mathbb{R}^{m-1}}(e^{200t_{n}})|}}

where |Bm1(e200tn)|subscript𝐵superscript𝑚1superscript𝑒200subscript𝑡𝑛|B_{\mathbb{R}^{m-1}}(e^{200t_{n}})| denotes the volume of

Bm1(e200tn)=NtnNsubscript𝐵superscript𝑚1superscript𝑒200subscript𝑡𝑛subscriptsuperscript𝑁subscript𝑡𝑛superscript𝑁B_{\mathbb{R}^{m-1}}(e^{200t_{n}})=N^{\prime}_{t_{n}}\subset N^{\prime}

with respect to the Euclidean parameters r𝑟r.

For each n𝑛n, let νnsubscript𝜈𝑛\nu_{n} denote the image of the measure μnsubscript𝜇𝑛\mu_{n} under the canonical projection from Mαsuperscript𝑀𝛼M^{\alpha} to G/Γ𝐺ΓG/\Gamma. The following proposition is shown in the next subsection.

Proposition 5.6.

There exists η>0𝜂0\eta>0 such that the sequence of measures {νn}subscript𝜈𝑛\{\nu_{n}\} has uniformly exponentially small mass in the cusp with exponent η𝜂\eta.

By the uniform comparability of distances in fibers of Mαsuperscript𝑀𝛼M^{\alpha}, this implies the family of measures {μn}subscript𝜇𝑛\{\mu_{n}\} has uniformly exponentially small measure in the cusp.

By Lemma 3.10(a) the families of measures {μn}subscript𝜇𝑛\{\mu_{n}\} and {νn}subscript𝜈𝑛\{\nu_{n}\} are precompact families. As Fnsubscript𝐹𝑛F_{n} is a Følner sequence in a solvable group, we have that any weak-* subsequential limit of {μn}subscript𝜇𝑛\{\mu_{n}\} or {νn}subscript𝜈𝑛\{\nu_{n}\} is AN𝐴superscript𝑁AN^{\prime}-invariant. Moreover, from Theorem 3.1(d), it follows that any weak-* subsequential limit νsubscript𝜈\nu_{\infty} of {νn}subscript𝜈𝑛\{\nu_{n}\} is invariant under the group Nsuperscript𝑁-N^{\prime} generated by the root groups Um,jsuperscript𝑈𝑚𝑗U^{m,j} for each 1jm11𝑗𝑚11\leq j\leq m-1. Since Nsuperscript𝑁N^{\prime} and Nsuperscript𝑁-N^{\prime} generate all of G𝐺G, we have that νsubscript𝜈\nu_{\infty} is a G𝐺G-invariant measure on G/Γ𝐺ΓG/\Gamma.

5.4. Proof of Proposition 5.6

5.4.1. Heuristics of the proof

The heuristic of the proof is the following. Observe that for a fixed choice of t𝑡t and sisubscript𝑠𝑖s_{i} as given by the choice of Følner set Fnsubscript𝐹𝑛F_{n}, the point

ati=1m3cisi(xn)superscript𝑎𝑡superscriptsubscriptproduct𝑖1𝑚3superscriptsubscript𝑐𝑖subscript𝑠𝑖subscript𝑥𝑛a^{t}\prod_{i=1}^{m-3}{c_{i}}^{s_{i}}(x_{n})

lies at sub-linear distance to the thick part of G/Γ𝐺ΓG/\Gamma with respect to tnsubscript𝑡𝑛t_{n}. Observe that the Nsuperscript𝑁N^{\prime}-orbit of such point is an embedded (m1)𝑚1(m-1)-dimensional torus in G/Γ𝐺ΓG/\Gamma. As the range of points in Nsuperscript𝑁N^{\prime} in the Følner set Fnsubscript𝐹𝑛F_{n} is quite large, averaging a Dirac measure of the point ati=1m3cisi(xn)superscript𝑎𝑡superscriptsubscriptproduct𝑖1𝑚3superscriptsubscript𝑐𝑖subscript𝑠𝑖subscript𝑥𝑛a^{t}\prod_{i=1}^{m-3}{c_{i}}^{s_{i}}(x_{n}) in the Nsuperscript𝑁N^{\prime}-direction in Fnsubscript𝐹𝑛F_{n} yields a measure quite close to Haar measure on the Nsuperscript𝑁N^{\prime}-orbit.

Observe that Nsuperscript𝑁N^{\prime}-orbits correspond to unstable manifolds for the action of the flow bssuperscript𝑏𝑠b^{s} on SL(m,)/SL(m,)SL𝑚SL𝑚\mathrm{SL}(m,\mathbb{R})/\mathrm{SL}(m,\mathbb{Z}). As the action of bssubscript𝑏𝑠b_{s} is known to be mixing, we expect that if s𝑠s is sufficiently large, flowing by bssubscript𝑏𝑠b_{s} the Nsuperscript𝑁N^{\prime}-orbit of ati=1m3cisi(xn)superscript𝑎𝑡superscriptsubscriptproduct𝑖1𝑚3superscriptsubscript𝑐𝑖subscript𝑠𝑖subscript𝑥𝑛a^{t}\prod_{i=1}^{m-3}{c_{i}}^{s_{i}}(x_{n}) will become equidistributed and in particular it will intersect non-trivially the thick part of G/Γ𝐺ΓG/\Gamma. This is the reason why the condition s>δ/2tn𝑠𝛿2subscript𝑡𝑛s>\delta/2t_{n} is assumed.

While intuition about mixing motivates the proof, we do not use it explicitly. Instead we use that for large enough s𝑠s, the action of bssubscript𝑏𝑠b_{s} expands the Nsuperscript𝑁N^{\prime}-orbits in a way that forces them to hit the thick part. We verify this fact by explicit matrix multiplication.

As bssuperscript𝑏𝑠b^{s} normalizes Nsuperscript𝑁N^{\prime}, the image under bssuperscript𝑏𝑠b^{s} of the Nsuperscript𝑁N^{\prime}-orbit of ati=1m3cisi(xn)superscript𝑎𝑡superscriptsubscriptproduct𝑖1𝑚3superscriptsubscript𝑐𝑖subscript𝑠𝑖subscript𝑥𝑛a^{t}\prod_{i=1}^{m-3}{c_{i}}^{s_{i}}(x_{n}) is the Nsuperscript𝑁N^{\prime}-orbit of a point ynsubscript𝑦𝑛y_{n} in the thick part of G/Γ𝐺ΓG/\Gamma. Having in mind the quantitative non-divergence of unipotent flows as in the proof Proposition 3.2, the Nsuperscript𝑁N^{\prime}-orbits have uniformly (over all n,si,𝑛subscript𝑠𝑖n,s_{i}, and t𝑡t) exponentially small mass in the cusps whence so do the measures νnsubscript𝜈𝑛\nu_{n}.

The following proof of Proposition 5.6 uses explicit matrix calculations and estimates to verify these heuristics.

5.4.2. Proof of Proposition 5.6

Recall that we identify each coset

gSL(m,)SL(m,)/SL(m,)𝑔SL𝑚SL𝑚SL𝑚g\mathrm{SL}(m,\mathbb{Z})\in\mathrm{SL}(m,\mathbb{R})/\mathrm{SL}(m,\mathbb{Z})

with a unimodular lattice Λg:=gmassignsubscriptΛ𝑔𝑔superscript𝑚\Lambda_{g}:=g\cdot\mathbb{Z}^{m} in msuperscript𝑚\mathbb{R}^{m}. We define the systole of a unimodular lattice ΛmΛsuperscript𝑚\Lambda\subset\mathbb{R}^{m} to be

δ(Λ):=minvΛ{0}vassign𝛿Λsubscript𝑣Λ0norm𝑣\delta(\Lambda):=\min_{v\in\Lambda\setminus\{0\}}\|v\|

and for an element gSL(m,)𝑔SL𝑚g\in\mathrm{SL}(m,\mathbb{R}), we denote by δ(g)𝛿𝑔\delta(g) the systole

δ(g)=δ(gm).𝛿𝑔𝛿𝑔superscript𝑚\delta(g)=\delta(g\cdot\mathbb{Z}^{m}).

From (9), to prove Proposition 5.6 it is sufficient to find η>0𝜂0\eta>0 so that the integrals

G/Γδ(g)η𝑑νn(gΓ)subscript𝐺Γ𝛿superscript𝑔𝜂differential-dsubscript𝜈𝑛𝑔Γ\int_{G/\Gamma}\delta(g)^{-\eta}\ d\nu_{n}(g\Gamma)

are uniformly bounded in n𝑛n.

As discussed in the above heuristic, from (32) to bound the integrals G/Γδη(g)𝑑νn(gΓ)subscript𝐺Γsuperscript𝛿𝜂𝑔differential-dsubscript𝜈𝑛𝑔Γ\int_{G/\Gamma}\delta^{-\eta}(g)\ d\nu_{n}(g\Gamma) it is sufficient to show each integral

1|B(e200tn)|B(e200tn)δ(atbs(Πcisi)urxn)η𝑑r1𝐵superscript𝑒200subscript𝑡𝑛subscript𝐵superscript𝑒200subscript𝑡𝑛𝛿superscriptsuperscript𝑎𝑡superscript𝑏𝑠Πsuperscriptsubscript𝑐𝑖subscript𝑠𝑖superscript𝑢𝑟subscript𝑥𝑛𝜂differential-d𝑟{\frac{1}{|B(e^{200t_{n}})|}}\int_{B(e^{200t_{n}})}\delta(a^{t}b^{s}(\Pi{c_{i}}^{s_{i}})u^{r}x_{n})^{-\eta}\ dr

is uniformly bounded in n𝑛n and in all parameters t,s,si𝑡𝑠subscript𝑠𝑖t,s,s_{i} for 0<t<tn,0𝑡subscript𝑡𝑛0<t<t_{n}, δtn/2<s<δtn,𝛿subscript𝑡𝑛2𝑠𝛿subscript𝑡𝑛\delta t_{n}/2<s<\delta t_{n}, and 0<si<tn.0subscript𝑠𝑖subscript𝑡𝑛0<s_{i}<\sqrt{t_{n}}. Recall here that xnG/Γsubscript𝑥𝑛𝐺Γx_{n}\in G/\Gamma are the points xnXthickH1,2/Λ1,2subscript𝑥𝑛subscript𝑋thicksubscript𝐻12subscriptΛ12x_{n}\in X_{\mathrm{thick}}\subset H_{1,2}/\Lambda_{1,2} satisfying (31) used in the construction of the measures μnsubscript𝜇𝑛\mu_{n}.

We have H1,2subscript𝐻12H_{1,2} is canonically embedded in SL(m,)SL𝑚\mathrm{SL}(m,\mathbb{R}). Given xnH1,2/Λ1,2subscript𝑥𝑛subscript𝐻12subscriptΛ12x_{n}\in H_{1,2}/\Lambda_{1,2}, let

x~nH1,2SL(m,)subscript~𝑥𝑛subscript𝐻12SL𝑚\tilde{x}_{n}\in H_{1,2}\subset\mathrm{SL}(m,\mathbb{R})

denote the element mapping to xnsubscript𝑥𝑛x_{n} under the map H1,2H1,2/Λ1,2subscript𝐻12subscript𝐻12subscriptΛ12H_{1,2}\to H_{1,2}/\Lambda_{1,2} which is contained in a fundamental domain contained in the Dirichlet domain 𝒟SL(2,)𝒟SL2\mathcal{D}\subset\mathrm{SL}(2,\mathbb{R}) in (7) in Section 2.2. Let \|\cdot\| denote the operator norm on SL(m,)SL𝑚\mathrm{SL}(m,\mathbb{R}) and m()𝑚m(\cdot) the associated conorm.

Claim 5.7.

For every n𝑛n, ttn𝑡subscript𝑡𝑛t\leq t_{n}, and 0sitn0subscript𝑠𝑖subscript𝑡𝑛0\leq s_{i}\leq\sqrt{t_{n}} as above, there exist

An=An,t,s1,,sm3SL(m1,)subscript𝐴𝑛subscript𝐴𝑛𝑡subscript𝑠1subscript𝑠𝑚3SL𝑚1A_{n}=A_{n,t,s_{1},\dots,s_{m-3}}\in\mathrm{SL}(m-1,\mathbb{R}) and γn=γn,t,s1,,sm3SL(m1,)subscript𝛾𝑛subscript𝛾𝑛𝑡subscript𝑠1subscript𝑠𝑚3SL𝑚1\gamma_{n}=\gamma_{n,t,s_{1},\dots,s_{m-3}}\in\mathrm{SL}(m-1,\mathbb{Z})

such that:

  1. (1)

    ati=1m3cisix~n=(Anγn0m1×101×m11)superscript𝑎𝑡superscriptsubscriptproduct𝑖1𝑚3superscriptsubscript𝑐𝑖subscript𝑠𝑖subscript~𝑥𝑛matrixsubscript𝐴𝑛subscript𝛾𝑛subscript0𝑚11subscript01𝑚11\displaystyle a^{t}\prod_{i=1}^{m-3}{c_{i}}^{s_{i}}\tilde{x}_{n}=\begin{pmatrix}A_{n}\gamma_{n}&0_{m-1\times 1}\\ 0_{1\times m-1}&1\end{pmatrix}

  2. (2)

    limnsupttn,0sitnlogAntn=0subscript𝑛subscriptsupremumformulae-sequence𝑡subscript𝑡𝑛0subscript𝑠𝑖subscript𝑡𝑛normsubscript𝐴𝑛subscript𝑡𝑛0\displaystyle\lim_{n\to\infty}\sup_{t\leq t_{n},0\leq s_{i}\leq\sqrt{t_{n}}}\frac{\log\|A_{n}\|}{t_{n}}=0 and limninfttn,0sitnlog(m(An))tn=0subscript𝑛subscriptinfimumformulae-sequence𝑡subscript𝑡𝑛0subscript𝑠𝑖subscript𝑡𝑛𝑚subscript𝐴𝑛subscript𝑡𝑛0\displaystyle\lim_{n\to\infty}\inf_{t\leq t_{n},0\leq s_{i}\leq\sqrt{t_{n}}}\frac{\log(m(A_{n}))}{t_{n}}=0

Proof.

(1) is immediate from construction. The uniform limit in (2) follows from Lemma 5.3(2), equation (5), and the fact that the sisubscript𝑠𝑖s_{i} are chosen so that 0sitn0subscript𝑠𝑖subscript𝑡𝑛0\leq s_{i}\leq\sqrt{t_{n}} whence

d(xn,at(Πcisi)xn)tn0𝑑subscript𝑥𝑛superscript𝑎𝑡Πsuperscriptsubscript𝑐𝑖subscript𝑠𝑖subscript𝑥𝑛subscript𝑡𝑛0\frac{d(x_{n},a^{t}(\Pi{c_{i}}^{s_{i}})\cdot x_{n})}{t_{n}}\to 0

uniformly in t,si𝑡subscript𝑠𝑖t,s_{i}. ∎

In the remainder, we will suppress the dependence of choices on t,s,si𝑡𝑠subscript𝑠𝑖t,s,s_{i}. We take KnSL(m1,)subscript𝐾𝑛SL𝑚1K_{n}\in\mathrm{SL}(m-1,\mathbb{R}) be such that

x~n=(Kn0m1×101×m11).subscript~𝑥𝑛matrixsubscript𝐾𝑛subscript0𝑚11subscript01𝑚11\tilde{x}_{n}=\begin{pmatrix}K_{n}&0_{m-1\times 1}\\ 0_{1\times m-1}&1\end{pmatrix}.

Note that Knsubscript𝐾𝑛K_{n} differs from the identity only in the first two rows and columns. Since each xnsubscript𝑥𝑛x_{n} is contained in Xthicksubscript𝑋thickX_{\mathrm{thick}}, we have that the matrix norm and conorm Knnormsubscript𝐾𝑛\|K_{n}\| and m(Kn)𝑚subscript𝐾𝑛m(K_{n}) are bounded above and below, respectively, by constants M1subscript𝑀1M_{1} and 1M11subscript𝑀1\frac{1}{M_{1}} independent of n𝑛n.

Recall r𝑟r denotes a vector in m1superscript𝑚1\mathbb{R}^{m-1} and urSL(m,)superscript𝑢𝑟SL𝑚u^{r}\in\mathrm{SL}(m,\mathbb{R}) is the unipotent element given by (29). Matrix computation yields

at(Πcisi)urx~n=(AnγnAnγnKn1r01×m11)superscript𝑎𝑡Πsuperscriptsubscript𝑐𝑖subscript𝑠𝑖superscript𝑢𝑟subscript~𝑥𝑛matrixsubscript𝐴𝑛subscript𝛾𝑛subscript𝐴𝑛subscript𝛾𝑛superscriptsubscript𝐾𝑛1𝑟subscript01𝑚11a^{t}(\Pi{c_{i}}^{s_{i}})u^{r}\tilde{x}_{n}=\begin{pmatrix}A_{n}\gamma_{n}&A_{n}\gamma_{n}K_{n}^{-1}r\\ 0_{1\times m-1}&1\end{pmatrix}

whence

bs(Πcisi)aturx~n=(esAnγnesAnγnKn1r01×m1e(m1)s).superscript𝑏𝑠Πsuperscriptsubscript𝑐𝑖subscript𝑠𝑖superscript𝑎𝑡superscript𝑢𝑟subscript~𝑥𝑛matrixsuperscript𝑒𝑠subscript𝐴𝑛subscript𝛾𝑛superscript𝑒𝑠subscript𝐴𝑛subscript𝛾𝑛superscriptsubscript𝐾𝑛1𝑟subscript01𝑚1superscript𝑒𝑚1𝑠b^{s}(\Pi{c_{i}}^{s_{i}})a^{t}u^{r}\tilde{x}_{n}=\begin{pmatrix}e^{s}A_{n}\gamma_{n}&e^{s}A_{n}\gamma_{n}K_{n}^{-1}r\\ 0_{1\times m-1}&e^{-(m-1)s}\end{pmatrix}.

We have

δ(bs(Πcisi)aturx~n)=δ(bs(Πcisi)aturxn)=infzm{0}(esAnγnesAnγnKn1r01×m1e(m1)s)z𝛿superscript𝑏𝑠Πsuperscriptsubscript𝑐𝑖subscript𝑠𝑖superscript𝑎𝑡superscript𝑢𝑟subscript~𝑥𝑛𝛿superscript𝑏𝑠Πsuperscriptsubscript𝑐𝑖subscript𝑠𝑖superscript𝑎𝑡superscript𝑢𝑟subscript𝑥𝑛subscriptinfimum𝑧superscript𝑚0delimited-∥∥matrixsuperscript𝑒𝑠subscript𝐴𝑛subscript𝛾𝑛superscript𝑒𝑠subscript𝐴𝑛subscript𝛾𝑛superscriptsubscript𝐾𝑛1𝑟subscript01𝑚1superscript𝑒𝑚1𝑠𝑧\begin{split}\delta(b^{s}(\Pi{c_{i}}^{s_{i}})a^{t}u^{r}\tilde{x}_{n})&=\delta(b^{s}(\Pi{c_{i}}^{s_{i}})a^{t}u^{r}x_{n})\\ &=\inf_{z\in\mathbb{Z}^{m}\setminus\{0\}}\bigg{\|}\begin{pmatrix}e^{s}A_{n}\gamma_{n}&e^{s}A_{n}\gamma_{n}K_{n}^{-1}r\\ 0_{1\times m-1}&e^{-(m-1)s}\end{pmatrix}z\bigg{\|}\end{split} (33)

To reduce notation, for fixed t,s,𝑡𝑠t,s, and sisubscript𝑠𝑖s_{i} define

β(r):=logδ(atbs(Πcisi)urx~n).assign𝛽𝑟𝛿superscript𝑎𝑡superscript𝑏𝑠Πsuperscriptsubscript𝑐𝑖subscript𝑠𝑖superscript𝑢𝑟subscript~𝑥𝑛\beta(r):=-\log\delta(a^{t}b^{s}(\Pi{c_{i}}^{s_{i}})u^{r}\tilde{x}_{n}).

We aim to find an upper bound of

1|B(e200tn)|B(e200tn)eηβ(r)𝑑r1𝐵superscript𝑒200subscript𝑡𝑛subscript𝐵superscript𝑒200subscript𝑡𝑛superscript𝑒𝜂𝛽𝑟differential-d𝑟\frac{1}{|B(e^{200t_{n}})|}\int_{B(e^{200t_{n}})}e^{\eta\beta(r)}dr

that is independent of n𝑛n and t,s,𝑡𝑠t,s, and sisubscript𝑠𝑖s_{i}.

Observe that if rr𝑟superscript𝑟r-{r^{\prime}} differ by an element of the unimodular lattice Knm1m1subscript𝐾𝑛superscript𝑚1superscript𝑚1K_{n}\mathbb{Z}^{m-1}\subset\mathbb{R}^{m-1}, then β(r)=β(r)𝛽𝑟𝛽superscript𝑟\beta(r)=\beta(r^{\prime}). Indeed, if r=r+Knzsuperscript𝑟𝑟subscript𝐾𝑛superscript𝑧r^{\prime}=r+K_{n}z^{\prime} for some z=(z1,,zm1)m1superscript𝑧superscriptsubscript𝑧1superscriptsubscript𝑧𝑚1superscript𝑚1z^{\prime}=(z_{1}^{\prime},\dots,z_{m-1}^{\prime})\in\mathbb{Z}^{m-1} and if zm{0}𝑧superscript𝑚0z\in\mathbb{Z}^{m}\smallsetminus\{0\} is z=(z1,,zm)𝑧subscript𝑧1subscript𝑧𝑚z=(z_{1},\dots,z_{m}) then

(esAnγnesAnγnKn1r01×m1e(m1)s)z=(esAnγnesAnγnKn1r01×m1e(m1)s)z~matrixsuperscript𝑒𝑠subscript𝐴𝑛subscript𝛾𝑛superscript𝑒𝑠subscript𝐴𝑛subscript𝛾𝑛superscriptsubscript𝐾𝑛1superscript𝑟subscript01𝑚1superscript𝑒𝑚1𝑠𝑧matrixsuperscript𝑒𝑠subscript𝐴𝑛subscript𝛾𝑛superscript𝑒𝑠subscript𝐴𝑛subscript𝛾𝑛superscriptsubscript𝐾𝑛1𝑟subscript01𝑚1superscript𝑒𝑚1𝑠~𝑧\begin{pmatrix}e^{s}A_{n}\gamma_{n}&e^{s}A_{n}\gamma_{n}K_{n}^{-1}r^{\prime}\\ 0_{1\times m-1}&e^{-(m-1)s}\end{pmatrix}z=\begin{pmatrix}e^{s}A_{n}\gamma_{n}&e^{s}A_{n}\gamma_{n}K_{n}^{-1}r\\ 0_{1\times m-1}&e^{-(m-1)s}\end{pmatrix}\tilde{z}

where z~=(z1+zmz1,,zm1+zmzm1,zm)m{0}.~𝑧subscript𝑧1subscript𝑧𝑚subscriptsuperscript𝑧1subscript𝑧𝑚1subscript𝑧𝑚subscriptsuperscript𝑧𝑚1subscript𝑧𝑚superscript𝑚0\tilde{z}=(z_{1}+z_{m}z^{\prime}_{1},\dots,z_{m-1}+z_{m}z^{\prime}_{m-1},z_{m})\in\mathbb{Z}^{m}\smallsetminus\{0\}. Thus we have that β:m1(0,):𝛽superscript𝑚10\beta\colon\mathbb{R}^{m-1}\to(0,\infty) descends to a function on the torus m1/(Knm1)superscript𝑚1subscript𝐾𝑛superscript𝑚1\mathbb{R}^{m-1}/(K_{n}\mathbb{Z}^{m-1}).

Let Dn=Kn([1/2,1/2]m1)subscript𝐷𝑛subscript𝐾𝑛superscript1212𝑚1D_{n}=K_{n}\cdot([-1/2,1/2]^{m-1}) be a fundamental domain for this torus in m1superscript𝑚1\mathbb{R}^{m-1} centered at 00. Let cnsubscript𝑐𝑛c_{n} denote the number of (Knm1)subscript𝐾𝑛superscript𝑚1(K_{n}\mathbb{Z}^{m-1})-translates of Dnsubscript𝐷𝑛D_{n} that intersect B(e200tn)𝐵superscript𝑒200subscript𝑡𝑛B(e^{200t_{n}}). Then, if tnsubscript𝑡𝑛t_{n} is sufficiently large we have that

1|B(e200tn)|B(e200tn)eηβ(r)𝑑r1|B(e200tn)|cnDneηβ(r)𝑑r2Dneηβ(r)𝑑r1𝐵superscript𝑒200subscript𝑡𝑛subscript𝐵superscript𝑒200subscript𝑡𝑛superscript𝑒𝜂𝛽𝑟differential-d𝑟1𝐵superscript𝑒200subscript𝑡𝑛subscript𝑐𝑛subscriptsubscript𝐷𝑛superscript𝑒𝜂𝛽𝑟differential-d𝑟2subscriptsubscript𝐷𝑛superscript𝑒𝜂𝛽𝑟differential-d𝑟\frac{1}{|B(e^{200t_{n}})|}\int_{B(e^{200t_{n}})}e^{\eta\beta(r)}\ dr\leq\frac{1}{|B(e^{200t_{n}})|}c_{n}\int_{D_{n}}e^{\eta\beta(r)}\ dr\leq 2\int_{D_{n}}e^{\eta\beta(r)}\ dr

The first inequality follows from inclusion. The second inequality follows from the fact that the perimeter of B(q)𝐵𝑞B(q) grows like qm2superscript𝑞𝑚2q^{m-2}, the volume of B(q)𝐵𝑞B(q) grows like qm1superscript𝑞𝑚1q^{m-1}, and the domains Dn=Kn([1/2,1/2]m1)subscript𝐷𝑛subscript𝐾𝑛superscript1212𝑚1D_{n}=K_{n}\cdot([-1/2,1/2]^{m-1}) have uniformly comparable geometry over n𝑛n.

It remains to estimate Dneηβ(r)𝑑rsubscriptsubscript𝐷𝑛superscript𝑒𝜂𝛽𝑟differential-d𝑟\int_{D_{n}}e^{\eta\beta(r)}\ dr. Given c>0𝑐0c>0 and fixed n,t,si,𝑛𝑡subscript𝑠𝑖n,t,s_{i}, and s𝑠s we define

Tc={rDn:β(r)>c}.subscript𝑇𝑐conditional-set𝑟subscript𝐷𝑛𝛽𝑟𝑐T_{c}=\{r\in D_{n}:\beta(r)>c\}.

Proposition 5.6 follows immediately from the estimate in the following lemma.

Lemma 5.8.

There exists constants M3,M4>0subscript𝑀3subscript𝑀40M_{3},M_{4}>0, independent of n,t,si,𝑛𝑡subscript𝑠𝑖n,t,s_{i}, and s𝑠s, such that

|Tc|M3ecM4.subscript𝑇𝑐subscript𝑀3superscript𝑒𝑐subscript𝑀4|T_{c}|\leq M_{3}e^{-cM_{4}}.

Indeed, if η1>M4superscript𝜂1subscript𝑀4\eta^{-1}>{M_{4}} then

Dneηβ(r)𝑑rsubscriptsubscript𝐷𝑛superscript𝑒𝜂𝛽𝑟differential-d𝑟\displaystyle\int_{D_{n}}e^{\eta\beta(r)}\ dr =0|{rDn:eηβ(r)τ}|𝑑τ1+1|{rDn:eηβ(r)τ}|𝑑τabsentsuperscriptsubscript0conditional-set𝑟subscript𝐷𝑛superscript𝑒𝜂𝛽𝑟𝜏differential-d𝜏1superscriptsubscript1conditional-set𝑟subscript𝐷𝑛superscript𝑒𝜂𝛽𝑟𝜏differential-d𝜏\displaystyle=\int_{0}^{\infty}|\{r\in D_{n}:e^{\eta\beta(r)}\geq\tau\}|\ d\tau\leq 1+\int_{1}^{\infty}|\{r\in D_{n}:e^{\eta\beta(r)}\geq\tau\}|\ d\tau
=1+1|{rDn:β(r)log(τ1η)}|𝑑τ=1+1|Tlog(τ1η)|𝑑τabsent1superscriptsubscript1conditional-set𝑟subscript𝐷𝑛𝛽𝑟superscript𝜏1𝜂differential-d𝜏1superscriptsubscript1subscript𝑇superscript𝜏1𝜂differential-d𝜏\displaystyle=1+\int_{1}^{\infty}|\{r\in D_{n}:\beta(r)\geq\log\big{(}\tau^{\frac{1}{\eta}}\big{)}\}|\ d\tau=1+\int_{1}^{\infty}|T_{\log\big{(}\tau^{\frac{1}{\eta}}\big{)}}|\ d\tau
=1+1M3τM4η𝑑τ<absent1superscriptsubscript1subscript𝑀3superscript𝜏subscript𝑀4𝜂differential-d𝜏\displaystyle=1+\int_{1}^{\infty}M_{3}\tau^{\frac{-M_{4}}{\eta}}\ d\tau<\infty

and Proposition 5.6 follows.

Proof of Lemma 5.8.

From (33), given any rm1𝑟superscript𝑚1r\in\mathbb{R}^{m-1}, if β(r)>c𝛽𝑟𝑐\beta(r)>c then there exists a non-zero z=(z1,z2,z3,.,zm)mz=(z_{1},z_{2},z_{3},....,z_{m})\in\mathbb{Z}^{m} such that

esAnγn(z1,,zm1)+zmAnγnKn1r<ec and |zm|<ece(m1)sformulae-sequencesuperscript𝑒𝑠normsubscript𝐴𝑛subscript𝛾𝑛subscript𝑧1subscript𝑧𝑚1subscript𝑧𝑚subscript𝐴𝑛subscript𝛾𝑛superscriptsubscript𝐾𝑛1𝑟superscript𝑒𝑐 and subscript𝑧𝑚superscript𝑒𝑐superscript𝑒𝑚1𝑠e^{s}\left\|A_{n}\gamma_{n}(z_{1},\dots,z_{m-1})+z_{m}A_{n}\gamma_{n}K_{n}^{-1}r\right\|<e^{-c}\ \ \text{ and }\ \ |z_{m}|<e^{-c}e^{(m-1)s}

which (as γnSL(m1,)subscript𝛾𝑛SL𝑚1\gamma_{n}\in\mathrm{SL}(m-1,\mathbb{Z})) holds if and only if there is a non-zero z=(z1,z2,z3,.,zm)mz=(z_{1},z_{2},z_{3},....,z_{m})\in\mathbb{Z}^{m}

esAn((z1,,zm1)+zmKn1(KnγnKn1)r)<ec and |zm|<ece(m1)sformulae-sequencesuperscript𝑒𝑠normsubscript𝐴𝑛subscript𝑧1subscript𝑧𝑚1subscript𝑧𝑚superscriptsubscript𝐾𝑛1subscript𝐾𝑛subscript𝛾𝑛superscriptsubscript𝐾𝑛1𝑟superscript𝑒𝑐 and subscript𝑧𝑚superscript𝑒𝑐superscript𝑒𝑚1𝑠e^{s}\left\|A_{n}\Big{(}(z_{1},\dots,z_{m-1})+z_{m}K_{n}^{-1}(K_{n}\gamma_{n}K_{n}^{-1})r\Big{)}\right\|<e^{-c}\ \ \text{ and }\ \ |z_{m}|<e^{-c}e^{(m-1)s} (34)

As KnγnKn1subscript𝐾𝑛subscript𝛾𝑛superscriptsubscript𝐾𝑛1K_{n}\gamma_{n}K_{n}^{-1} induces a volume-preserving automorphism of m1/(Knm1)superscript𝑚1subscript𝐾𝑛superscript𝑚1\mathbb{R}^{m-1}/(K_{n}\mathbb{Z}^{m-1}), the set of rDn𝑟subscript𝐷𝑛r\in D_{n} satisfying (34) for some zm𝑧superscript𝑚z\in\mathbb{Z}^{m} has the same measure as the set of rDn𝑟subscript𝐷𝑛r\in D_{n} satisfying

esAn((z1,,zm1)+zmKn1r)<ec and |zm|<ece(m1)sformulae-sequencesuperscript𝑒𝑠normsubscript𝐴𝑛subscript𝑧1subscript𝑧𝑚1subscript𝑧𝑚superscriptsubscript𝐾𝑛1𝑟superscript𝑒𝑐 and subscript𝑧𝑚superscript𝑒𝑐superscript𝑒𝑚1𝑠e^{s}\left\|A_{n}\Big{(}(z_{1},\dots,z_{m-1})+z_{m}K_{n}^{-1}r\Big{)}\right\|<e^{-c}\ \ \text{ and }\ \ |z_{m}|<e^{-c}e^{(m-1)s}

for some zm.𝑧superscript𝑚z\in\mathbb{Z}^{m}.

For every integer k𝑘k satisfying |k|<ece(m1)s𝑘superscript𝑒𝑐superscript𝑒𝑚1𝑠|k|<e^{-c}e^{(m-1)s}, let Tc,ksubscript𝑇𝑐𝑘T_{c,k} be the subset of rDn𝑟subscript𝐷𝑛r\in D_{n} such that there exists (z1,z2,,zm1)m1subscript𝑧1subscript𝑧2subscript𝑧𝑚1superscript𝑚1(z_{1},z_{2},\dots,z_{m-1})\in\mathbb{Z}^{m-1} satisfying

esAn((z1,,zm1)+kKn1r)<ec.superscript𝑒𝑠normsubscript𝐴𝑛subscript𝑧1subscript𝑧𝑚1𝑘superscriptsubscript𝐾𝑛1𝑟superscript𝑒𝑐e^{s}\left\|A_{n}\Big{(}(z_{1},\dots,z_{m-1})+kK_{n}^{-1}r\Big{)}\right\|<e^{-c}.

Then |Tc||k|<ece(m1)s|Tc,k|.subscript𝑇𝑐subscript𝑘superscript𝑒𝑐superscript𝑒𝑚1𝑠subscript𝑇𝑐𝑘|T_{c}|\leq\sum_{|k|<e^{-c}e^{(m-1)s}}|T_{c,k}|. Thus the estimate reduces to the following.

Claim 5.9.

There exists M50subscript𝑀50M_{5}\geq 0 such that |Tc,k|<M5e(m1)(s+c)subscript𝑇𝑐𝑘subscript𝑀5superscript𝑒𝑚1𝑠𝑐|T_{c,k}|<M_{5}e^{-(m-1)(s+c)} for all n𝑛n sufficiently large.

Proof.

Recall that δtn/2<s𝛿subscript𝑡𝑛2𝑠\delta t_{n}/2<s. If k=0𝑘0k=0 then, for any non-zero (z1,,zm1)m1subscript𝑧1subscript𝑧𝑚1superscript𝑚1(z_{1},\dots,z_{m-1})\in\mathbb{Z}^{m-1}, we have

esAn(z1,,zm1)>eδtn/2m(An).superscript𝑒𝑠normsubscript𝐴𝑛subscript𝑧1subscript𝑧𝑚1superscript𝑒𝛿subscript𝑡𝑛2𝑚subscript𝐴𝑛e^{s}\left\|A_{n}(z_{1},\dots,z_{m-1})\right\|>e^{\delta t_{n}/2}m(A_{n}).

From Claim 5.7(2), if n𝑛n is large enough then m(An)eδtn/4𝑚subscript𝐴𝑛superscript𝑒𝛿subscript𝑡𝑛4m(A_{n})\geq e^{-\delta t_{n}/4} and so the term in the left hand side above is greater than one, therefore Tc,0=subscript𝑇𝑐0T_{c,0}=\varnothing for n𝑛n sufficiently large.

If k0𝑘0k\neq 0, observe that the map Mk:m1/Knm1m1/Knm1:subscript𝑀𝑘superscript𝑚1subscript𝐾𝑛superscript𝑚1superscript𝑚1subscript𝐾𝑛superscript𝑚1M_{k}\colon\mathbb{R}^{m-1}/K_{n}\mathbb{Z}^{m-1}\to\mathbb{R}^{m-1}/K_{n}\mathbb{Z}^{m-1} given by

r+Knm1kr+Knm1maps-to𝑟subscript𝐾𝑛superscript𝑚1𝑘𝑟subscript𝐾𝑛superscript𝑚1r+K_{n}\mathbb{Z}^{m-1}\mapsto kr+K_{n}\mathbb{Z}^{m-1}

preserves the Lebesgue measure on m1/Knm1.superscript𝑚1subscript𝐾𝑛superscript𝑚1\mathbb{R}^{m-1}/K_{n}\mathbb{Z}^{m-1}. In particular, this implies that Tc,ksubscript𝑇𝑐𝑘T_{c,k} and Tc,1subscript𝑇𝑐1T_{c,1} have the same volume.

We thus take k=1𝑘1k=1. Note that Kn1Dn=[1/2,1/2]superscriptsubscript𝐾𝑛1subscript𝐷𝑛1212K_{n}^{-1}D_{n}=[-1/2,1/2]. There is a L1𝐿1L\geq 1, depending only on m1𝑚1m-1, such that the set

Q={zm1:|z+r|1 for some rKn1Dn }𝑄conditional-setsuperscript𝑧superscript𝑚1superscript𝑧𝑟1 for some rKn1Dn Q=\{z^{\prime}\in\mathbb{Z}^{m-1}:|z^{\prime}+r|\leq 1\text{ for some $r\in K_{n}^{-1}D_{n}$ }\}

has cardinality at most L𝐿L. From Claim 5.7(2), if n𝑛n is large enough then m(An)eδtn/4𝑚subscript𝐴𝑛superscript𝑒𝛿subscript𝑡𝑛4m(A_{n})\geq e^{-\delta t_{n}/4} whence for all (z1,,zm1)m1Qsubscript𝑧1subscript𝑧𝑚1superscript𝑚1𝑄(z_{1},\dots,z_{m-1})\in\mathbb{Z}^{m-1}\smallsetminus Q and all rDn𝑟subscript𝐷𝑛r\in D_{n},

esAn((z1,,zm1)+Kn1r)1.superscript𝑒𝑠normsubscript𝐴𝑛subscript𝑧1subscript𝑧𝑚1superscriptsubscript𝐾𝑛1𝑟1e^{s}\left\|A_{n}\Big{(}(z_{1},\dots,z_{m-1})+K_{n}^{-1}r\Big{)}\right\|\geq 1.

We thus need only consider (z1,,zm1)Qsubscript𝑧1subscript𝑧𝑚1𝑄(z_{1},\dots,z_{m-1})\in Q.

Given a fixed z=(z1,,zm1)Qm1𝑧subscript𝑧1subscript𝑧𝑚1𝑄superscript𝑚1z=(z_{1},\dots,z_{m-1})\in Q\subset\mathbb{Z}^{m-1}, using that KnSL(m1,)subscript𝐾𝑛SL𝑚1K_{n}\in\mathrm{SL}(m-1,\mathbb{R}) we have

|{rm1:z+Kn1r}|(2)m1conditional-set𝑟superscript𝑚1norm𝑧superscriptsubscript𝐾𝑛1𝑟superscript2𝑚1\left|\{r\in\mathbb{R}^{m-1}:\|z+K_{n}^{-1}r\|\leq\ell\}\right|\leq(2\ell)^{m-1}

whence

|{rm1:(z1,,zm1)+Kn1rec}|2m1ec(m1).conditional-set𝑟superscript𝑚1normsubscript𝑧1subscript𝑧𝑚1superscriptsubscript𝐾𝑛1𝑟superscript𝑒𝑐superscript2𝑚1superscript𝑒𝑐𝑚1\left|\{r\in\mathbb{R}^{m-1}:\|(z_{1},\dots,z_{m-1})+K_{n}^{-1}r\|\leq e^{-c}\}\right|\leq 2^{m-1}e^{-c(m-1)}.

If rTc,1𝑟subscript𝑇𝑐1r\in T_{c,1} so that

esAn((z1,,zm1)+Kn1r)ecsuperscript𝑒𝑠normsubscript𝐴𝑛subscript𝑧1subscript𝑧𝑚1superscriptsubscript𝐾𝑛1𝑟superscript𝑒𝑐e^{s}\left\|A_{n}\Big{(}(z_{1},\dots,z_{m-1})+K_{n}^{-1}r\Big{)}\right\|\leq e^{-c}

then

An((z1,,zm1)+Kn1r)ecs.normsubscript𝐴𝑛subscript𝑧1subscript𝑧𝑚1superscriptsubscript𝐾𝑛1𝑟superscript𝑒𝑐𝑠\left\|A_{n}\Big{(}(z_{1},\dots,z_{m-1})+K_{n}^{-1}r\Big{)}\right\|\leq e^{-c-s}. (35)

Since AnSL(m1,)subscript𝐴𝑛SL𝑚1A_{n}\in\mathrm{SL}(m-1,\mathbb{R}) the set of rm1𝑟superscript𝑚1r\in\mathbb{R}^{m-1} satisfying (35) has the same volume as the set of rm1𝑟superscript𝑚1r\in\mathbb{R}^{m-1} satisfying

(z1,,zm1)+Kn1recs.normsubscript𝑧1subscript𝑧𝑚1superscriptsubscript𝐾𝑛1𝑟superscript𝑒𝑐𝑠\left\|(z_{1},\dots,z_{m-1})+K_{n}^{-1}r\right\|\leq e^{-c-s}.

It follows that |Tc,1|2m1Le(s+c)(m1)subscript𝑇𝑐1superscript2𝑚1𝐿superscript𝑒𝑠𝑐𝑚1|T_{c,1}|\leq 2^{m-1}Le^{-(s+c)(m-1)}.∎

To finish the proof of Lemma 5.8, from Claim 5.9 we have

|Tc||k|<ece(m1)s|Tc,k|(2ece(m1)s+1)M5e(m1)(s+c)M3ecM4subscript𝑇𝑐subscript𝑘superscript𝑒𝑐superscript𝑒𝑚1𝑠subscript𝑇𝑐𝑘2superscript𝑒𝑐superscript𝑒𝑚1𝑠1subscript𝑀5superscript𝑒𝑚1𝑠𝑐subscript𝑀3superscript𝑒𝑐subscript𝑀4|T_{c}|\leq\sum_{|k|<e^{-c}e^{(m-1)s}}|T_{c,k}|\leq(2e^{-c}e^{(m-1)s}+1)M_{5}e^{-(m-1)(s+c)}\leq M_{3}e^{-cM_{4}}

for some constants M3,M4subscript𝑀3subscript𝑀4M_{3},M_{4} independent of n𝑛n. ∎

5.5. Positive Lyapunov exponents for limit measures

To deduce Proposition 5.1, having assumed that χmaxsubscript𝜒\chi_{\max} in (25) is non-zero, we show that any weak-{*} subsequential limit of the sequence of measures {μn}subscript𝜇𝑛\{\mu_{n}\} has a positive Lyapunov exponent from which we derive a contradiction.

Recall from Section 5.3 that we fixed sequences xn,vn,tnsubscript𝑥𝑛subscript𝑣𝑛subscript𝑡𝑛x_{n},v_{n},t_{n} such that Dxnatn(vn)eλtnnormsubscript𝐷subscript𝑥𝑛superscript𝑎subscript𝑡𝑛subscript𝑣𝑛superscript𝑒𝜆subscript𝑡𝑛\|D_{x_{n}}a^{t_{n}}(v_{n})\|\geq e^{\lambda t_{n}} for some fixed λ>0𝜆0\lambda>0. Let 𝒜:G×FF:𝒜𝐺𝐹𝐹\mathcal{A}\colon G\times F\to F be the fiberwise derivative cocycle over the action of G𝐺G on Mαsuperscript𝑀𝛼M^{\alpha}.

Our main result is the following.

Proposition 5.10.

For any weak-* subsequential limit μsubscript𝜇\mu_{\infty} of {μn}subscript𝜇𝑛\{\mu_{n}\} we have

λtop,a,μ,𝒜λ/2>0.subscript𝜆top𝑎subscript𝜇𝒜𝜆20\lambda_{\mathrm{top},a,\mu_{\infty},\mathcal{A}}\geq\lambda/2>0.

We first show that averaging over Nsuperscript𝑁N^{\prime} does not change the Lyapunov exponents of the cocycle.

Claim 5.11.

Given any ε>0𝜀0\varepsilon>0 there is tε>0subscript𝑡𝜀0t_{\varepsilon}>0 such that for any ttε𝑡subscript𝑡𝜀t\geq t_{\varepsilon} and any rBm1(et)𝑟subscript𝐵superscript𝑚1superscript𝑒𝑡r\in B_{\mathbb{R}^{m-1}}(e^{t}) we have

DxurFibereεtsubscriptnormsubscript𝐷𝑥superscript𝑢𝑟Fibersuperscript𝑒𝜀𝑡\|{D_{x}u^{r}}\|_{\mathrm{Fiber}}\leq e^{\varepsilon t}

for any xXthick𝑥subscript𝑋thickx\in X_{\mathrm{thick}}.

Proof.

Recall that the Nsuperscript𝑁N^{\prime}-orbit of any xX:=H1,2/Λ1,2SL(m,)/SL(m,)𝑥𝑋assignsubscript𝐻12subscriptΛ12SL𝑚SL𝑚x\in X:=H_{1,2}/\Lambda_{1,2}\subset\mathrm{SL}(m,\mathbb{R})/\mathrm{SL}(m,\mathbb{Z}) is a closed torus. Then the Nsuperscript𝑁N^{\prime}-orbit of Xthicksubscript𝑋thickX_{\mathrm{thick}} is compact. Recall our fixed fundamental domain 𝒟~~𝒟\mathcal{F}\subset\widetilde{\mathcal{D}} contained in the Dirichlet domain 𝒟~~𝒟\widetilde{\mathcal{D}} of the identity for SL(m,)/SL(m,)SL𝑚SL𝑚\mathrm{SL}(m,\mathbb{R})/\mathrm{SL}(m,\mathbb{Z}) as discussed in Section 2.2. Given xSL(m,)/SL(m,)𝑥SL𝑚SL𝑚x\in\mathrm{SL}(m,\mathbb{R})/\mathrm{SL}(m,\mathbb{Z}), let x~~𝑥\tilde{x} be the lift of x𝑥x in \mathcal{F}. Let X~thickH1,2subscript~𝑋thicksubscript𝐻12\widetilde{X}_{\mathrm{thick}}\subset H_{1,2}\cap\mathcal{F} denote the lift of Xthicksubscript𝑋thickX_{\mathrm{thick}} to \mathcal{F} and let X^thicksubscript^𝑋thick\hat{X}_{\mathrm{thick}} be the lift of the orbit NXthicksuperscript𝑁subscript𝑋thickN^{\prime}X_{\mathrm{thick}} to \mathcal{F}. As discussed in Section 2.2, we have that X~thicksubscript~𝑋thick\widetilde{X}_{\mathrm{thick}} is contained in the Dirichlet domain 𝒟𝒟\mathcal{D} of the identity for the Λ1,2subscriptΛ12\Lambda_{1,2}-action on H1,2subscript𝐻12H_{1,2}. Moreover, X^thicksubscript^𝑋thick\hat{X}_{\mathrm{thick}} is precompact in SL(m,)SL𝑚\mathrm{SL}(m,\mathbb{R}).

Fix rm1𝑟superscript𝑚1r\in\mathbb{R}^{m-1} and xXthick𝑥subscript𝑋thickx\in X_{\mathrm{thick}}. Write

x~=(K001)~𝑥𝐾001\tilde{x}=\left(\begin{array}[]{cc}K&0\\ 0&1\end{array}\right)

for some KSL(m1,)𝐾SL𝑚1K\in\mathrm{SL}(m-1,\mathbb{R}); we have KM1norm𝐾subscript𝑀1\|K\|\leq M_{1} and m(K)1M1𝑚𝐾1subscript𝑀1m(K)\geq\frac{1}{M_{1}} for all xXthick𝑥subscript𝑋thickx\in X_{\mathrm{thick}}. The deck group of the orbit Nx~superscript𝑁~𝑥N^{\prime}\tilde{x} is

x~{uz:zm1}x~1={uKz:zm1}.~𝑥conditional-setsuperscript𝑢𝑧𝑧superscript𝑚1superscript~𝑥1conditional-setsuperscript𝑢𝐾𝑧𝑧superscript𝑚1\tilde{x}\{u^{z}:z\in\mathbb{Z}^{m-1}\}\tilde{x}^{-1}=\{u^{K\cdot z}:z\in\mathbb{Z}^{m-1}\}.

Thus, there is zm1𝑧superscript𝑚1z\in\mathbb{Z}^{m-1} and rm1superscript𝑟superscript𝑚1r^{\prime}\in\mathbb{R}^{m-1} such that

urx~=(Kr01)=(Kr+Kz01)=(1r01)(K001)(1z01)=urx~uzsuperscript𝑢𝑟~𝑥𝐾𝑟01𝐾superscript𝑟𝐾𝑧011superscript𝑟01𝐾0011𝑧01superscript𝑢superscript𝑟~𝑥superscript𝑢𝑧u^{r}\tilde{x}=\left(\begin{array}[]{cc}K&r\\ 0&1\end{array}\right)=\left(\begin{array}[]{cc}K&r^{\prime}+Kz\\ 0&1\end{array}\right)=\left(\begin{array}[]{cc}1&r^{\prime}\\ 0&1\end{array}\right)\left(\begin{array}[]{cc}K&0\\ 0&1\end{array}\right)\left(\begin{array}[]{cc}1&z\\ 0&1\end{array}\right)=u^{r^{\prime}}\tilde{x}u^{z}

and urx~X^thicksuperscript𝑢superscript𝑟~𝑥subscript^𝑋thicku^{r^{\prime}}\tilde{x}\in\hat{X}_{\mathrm{thick}}. Then

DxurDxx~1FiberDIdΓuzFiberDIdΓurx~Fiber.normsubscript𝐷𝑥superscript𝑢𝑟subscriptnormsubscript𝐷𝑥superscript~𝑥1Fibersubscriptnormsubscript𝐷IdΓsuperscript𝑢𝑧Fibersubscriptnormsubscript𝐷IdΓsuperscript𝑢superscript𝑟~𝑥Fiber\|D_{x}u^{r}\|\leq\|D_{x}\tilde{x}^{-1}\|_{\mathrm{Fiber}}\cdot\|D_{\operatorname{Id}\Gamma}u^{z}\|_{\mathrm{Fiber}}\cdot\|D_{\operatorname{Id}\Gamma}u^{r^{\prime}}\tilde{x}\|_{\mathrm{Fiber}}.

Since x~~𝑥\tilde{x} and urx~superscript𝑢superscript𝑟~𝑥u^{r^{\prime}}\tilde{x} are in precompact sets, the first and last terms of the right hand side are uniformly bounded in r𝑟r and xXthick𝑥subscript𝑋thickx\in X_{\mathrm{thick}}.

There exists some C𝐶C such that

DIdΓuzFiberCDα(uz).subscriptnormsubscript𝐷IdΓsuperscript𝑢𝑧Fiber𝐶norm𝐷𝛼superscript𝑢𝑧\|D_{\operatorname{Id}\Gamma}u^{z}\|_{\mathrm{Fiber}}\leq C\|D\alpha(u^{z})\|.

Since rBm1(et)𝑟subscript𝐵superscript𝑚1superscript𝑒𝑡r\in B_{\mathbb{R}^{m-1}}(e^{t}) we have zBm1(M1et)𝑧subscript𝐵superscript𝑚1subscript𝑀1superscript𝑒𝑡z\in B_{\mathbb{R}^{m-1}}(M_{1}e^{t}) whence d(uz,Id)C2t+C3𝑑superscript𝑢𝑧Idsubscript𝐶2𝑡subscript𝐶3d(u^{z},\mathrm{Id})\leq C_{2}t+C_{3} for some constants C2subscript𝐶2C_{2} and C3subscript𝐶3C_{3}. Proposition 4.1 implies for any εsuperscript𝜀\varepsilon^{\prime} that

Dα(uz)eε(C2t+C3)norm𝐷𝛼superscript𝑢𝑧superscript𝑒superscript𝜀subscript𝐶2𝑡subscript𝐶3\|D\alpha(u^{z})\|\leq e^{\varepsilon^{\prime}(C_{2}t+C_{3})}

and taking ε>0superscript𝜀0\varepsilon^{\prime}>0 sufficiently small, the claim follows. ∎

By Lemma 2.1, the fact that SL(m,)SL𝑚\mathrm{SL}(m,\mathbb{Z}) is finitely generated, and the uniform comparability of the fibers of Mαsuperscript𝑀𝛼M^{\alpha}, we also have the following.

Claim 5.12.

There are uniform constants C5subscript𝐶5C_{5} and C6subscript𝐶6C_{6} with the following property: Let xG/Γ𝑥𝐺Γx\in G/\Gamma. Then for any X𝔤𝑋𝔤X\in\mathfrak{g} with X1norm𝑋1\|X\|\leq 1 we have

(Dxexp(tX))±1FibereC5t+C5d(x,Id)+C6.subscriptnormsuperscriptsubscript𝐷𝑥𝑡𝑋plus-or-minus1Fibersuperscript𝑒subscript𝐶5𝑡subscript𝐶5𝑑𝑥Idsubscript𝐶6\left\|\big{(}D_{x}\exp(tX)\big{)}^{\pm 1}\right\|_{\mathrm{Fiber}}\leq e^{C_{5}t+C_{5}d(x,\mathrm{Id})+C_{6}}.

We now prove Proposition 5.10.

Proof of Proposition 5.10.

Recall we take xnXthicksubscript𝑥𝑛subscript𝑋thickx_{n}\in X_{\mathrm{thick}}, tnsubscript𝑡𝑛t_{n}\to\infty, and vnF(xn)subscript𝑣𝑛𝐹subscript𝑥𝑛v_{n}\in F(x_{n}) with vn=1normsubscript𝑣𝑛1\|v_{n}\|=1 such that Dxnatn(vn)eλtnnormsubscript𝐷subscript𝑥𝑛superscript𝑎subscript𝑡𝑛subscript𝑣𝑛superscript𝑒𝜆subscript𝑡𝑛\|D_{x_{n}}a^{t_{n}}(v_{n})\|\geq e^{\lambda t_{n}} for some fixed λ>0𝜆0\lambda>0 in (31) in Section 5.3. We also write 𝒜:G×FF:𝒜𝐺𝐹𝐹\mathcal{A}\colon G\times F\to F for the fiberwise derivative cocycle.

The measures μnsubscript𝜇𝑛\mu_{n} constructed in Section 5.3 are defined by averaging last along the orbit at,0ttnsuperscript𝑎𝑡0𝑡subscript𝑡𝑛a^{t},0\leq t\leq t_{n}. Let ξnsubscript𝜉𝑛\xi_{n} be the measure on Mαsuperscript𝑀𝛼M^{\alpha} given by

Mαsubscriptsuperscript𝑀𝛼\displaystyle\int_{M^{\alpha}} fdξn=2δtn(1tn)m31|Bm1(e200tn)|𝑓𝑑subscript𝜉𝑛2𝛿subscript𝑡𝑛superscript1subscript𝑡𝑛𝑚31subscript𝐵superscript𝑚1superscript𝑒200subscript𝑡𝑛\displaystyle f\ d\xi_{n}={\frac{2}{\delta t_{n}}\bigg{(}\frac{1}{\sqrt{t_{n}}}}\bigg{)}^{m-3}\frac{1}{|B_{\mathbb{R}^{m-1}}(e^{200t_{n}})|}
δtn/2δtn[0,tn]m3Bm1(e200tn)f(atbsc=1m3cisiur(xn,p(vn)))𝑑r𝑑si𝑑s.superscriptsubscript𝛿subscript𝑡𝑛2𝛿subscript𝑡𝑛subscriptsuperscript0subscript𝑡𝑛𝑚3subscriptsubscript𝐵superscript𝑚1superscript𝑒200subscript𝑡𝑛𝑓superscript𝑎𝑡superscript𝑏𝑠superscriptsubscriptproduct𝑐1𝑚3superscriptsubscript𝑐𝑖subscript𝑠𝑖superscript𝑢𝑟subscript𝑥𝑛𝑝subscript𝑣𝑛differential-d𝑟differential-dsubscript𝑠𝑖differential-d𝑠\displaystyle\quad\quad\int_{\delta t_{n}/2}^{\delta t_{n}}\int_{[0,\sqrt{t_{n}}]^{m-3}}\int_{B_{\mathbb{R}^{m-1}}(e^{200t_{n}})}f\left(a^{t}b^{s}\prod_{c=1}^{m-3}c_{i}^{s_{i}}u^{r}\cdot(x_{n},p(v_{n}))\right)\ dr\ d{s_{i}}\ ds.

In the context of Lemma 3.10, the measures μn=0tn(atξn)𝑑tsubscript𝜇𝑛superscriptsubscript0subscript𝑡𝑛subscriptsuperscript𝑎𝑡subscript𝜉𝑛differential-d𝑡\mu_{n}=\int_{0}^{t_{n}}(a^{t}_{*}\xi_{n})\ dt constructed in Section 5.3 correspond to the empirical measures ηn=η(loga,tn,ξn)subscript𝜂𝑛𝜂𝑎subscript𝑡𝑛subscript𝜉𝑛\eta_{n}=\eta(\log a,t_{n},\xi_{n}) appearing in the proof of Lemma 3.10. From Lemma 3.10, to establish Proposition 5.10 it is sufficient to show that

log𝒜(atn,)dξnλ2tn.norm𝒜superscript𝑎subscript𝑡𝑛𝑑subscript𝜉𝑛𝜆2subscript𝑡𝑛\int\log\|\mathcal{A}(a^{t_{n}},\cdot)\|\ d\xi_{n}\geq\frac{\lambda}{2}t_{n}.

We have

Mαsubscriptsuperscript𝑀𝛼\displaystyle\int_{M^{\alpha}} log𝒜(atn,)dξnnorm𝒜superscript𝑎subscript𝑡𝑛𝑑subscript𝜉𝑛\displaystyle{\log\|\mathcal{A}(a^{t_{n}},\cdot)\|\ d\xi_{n}}
=2δtn(1tn)m31|Bm1(e200tn)|absent2𝛿subscript𝑡𝑛superscript1subscript𝑡𝑛𝑚31subscript𝐵superscript𝑚1superscript𝑒200subscript𝑡𝑛\displaystyle\quad={\frac{2}{\delta t_{n}}\bigg{(}\frac{1}{\sqrt{t_{n}}}}\bigg{)}^{m-3}\frac{1}{|B_{\mathbb{R}^{m-1}}(e^{200t_{n}})|}
δtn/2δtn[0,tn]m3Bm1(e200tn)log𝒜(atn,bsΠcisiur(xn,p(vn)))drdsidssuperscriptsubscript𝛿subscript𝑡𝑛2𝛿subscript𝑡𝑛subscriptsuperscript0subscript𝑡𝑛𝑚3subscriptsubscript𝐵superscript𝑚1superscript𝑒200subscript𝑡𝑛norm𝒜superscript𝑎subscript𝑡𝑛superscript𝑏𝑠Πsuperscriptsubscript𝑐𝑖subscript𝑠𝑖superscript𝑢𝑟subscript𝑥𝑛𝑝subscript𝑣𝑛𝑑𝑟𝑑subscript𝑠𝑖𝑑𝑠\displaystyle\quad\quad\int_{\delta t_{n}/2}^{\delta t_{n}}\int_{[0,\sqrt{t_{n}}]^{m-3}}\int_{B_{\mathbb{R}^{m-1}}(e^{200t_{n}})}{\log\left\|\mathcal{A}\big{(}a^{t_{n}},b^{s}\Pi c_{i}^{s_{i}}u^{r}\cdot(x_{n},p(v_{n}))\big{)}\right\|\ dr\ ds_{i}\ ds}
2δtn(1tn)m31|Bm1(e200tn)|absent2𝛿subscript𝑡𝑛superscript1subscript𝑡𝑛𝑚31subscript𝐵superscript𝑚1superscript𝑒200subscript𝑡𝑛\displaystyle\quad\geq{\frac{2}{\delta t_{n}}\bigg{(}\frac{1}{\sqrt{t_{n}}}}\bigg{)}^{m-3}\frac{1}{|B_{\mathbb{R}^{m-1}}(e^{200t_{n}})|}
δtn/2δtn[0,tn]m3Bm1(e200tn)logDxn(atnbsΠcisiur)(vn)Dxn(bsΠcisiur)(vn)drdsids.superscriptsubscript𝛿subscript𝑡𝑛2𝛿subscript𝑡𝑛subscriptsuperscript0subscript𝑡𝑛𝑚3subscriptsubscript𝐵superscript𝑚1superscript𝑒200subscript𝑡𝑛normsubscript𝐷subscript𝑥𝑛superscript𝑎subscript𝑡𝑛superscript𝑏𝑠Πsuperscriptsubscript𝑐𝑖subscript𝑠𝑖superscript𝑢𝑟subscript𝑣𝑛normsubscript𝐷subscript𝑥𝑛superscript𝑏𝑠Πsuperscriptsubscript𝑐𝑖subscript𝑠𝑖superscript𝑢𝑟subscript𝑣𝑛𝑑𝑟𝑑subscript𝑠𝑖𝑑𝑠\displaystyle\quad\quad\int_{\delta t_{n}/2}^{\delta t_{n}}\int_{[0,\sqrt{t_{n}}]^{m-3}}\int_{B_{\mathbb{R}^{m-1}}(e^{200t_{n}})}\log\frac{\left\|D_{x_{n}}\big{(}a^{t_{n}}b^{s}\Pi c_{i}^{s_{i}}u^{r}\big{)}(v_{n})\right\|}{\left\|D_{x_{n}}\big{(}b^{s}\Pi c_{i}^{s_{i}}u^{r}\big{)}(v_{n})\right\|}\ dr\ ds_{i}\ ds.

Consider fixed r,𝑟r, s,𝑠s, and sisubscript𝑠𝑖s_{i}. Take rm1superscript𝑟superscript𝑚1r^{\prime}\in\mathbb{R}^{m-1} such that atnur=uratnsuperscript𝑎subscript𝑡𝑛superscript𝑢𝑟superscript𝑢superscript𝑟superscript𝑎subscript𝑡𝑛a^{t_{n}}u^{r}=u^{r^{\prime}}a^{t_{n}}. Then

log\displaystyle\log Dxn(atnbsΠcisiur)(vn)Dxn(bsΠcisiur)(vn))=logDatnxn(bsΠcisiur)Dxnatn(vn)Dxn(bsΠcisiur)(vn)\displaystyle\frac{\left\|D_{x_{n}}\big{(}a^{t_{n}}b^{s}\Pi c_{i}^{s_{i}}u^{r}\big{)}(v_{n})\right\|}{\left\|D_{x_{n}}\big{(}b^{s}\Pi c_{i}^{s_{i}}u^{r}\big{)}(v_{n})\big{)}\right\|}=\log\frac{\left\|D_{a^{t_{n}}\cdot x_{n}}\big{(}b^{s}\Pi c_{i}^{s_{i}}u^{r^{\prime}}\big{)}\circ D_{x_{n}}a^{t_{n}}\big{(}v_{n}\big{)}\right\|}{\left\|D_{x_{n}}\big{(}b^{s}\Pi c_{i}^{s_{i}}u^{r}\big{)}(v_{n})\right\|}
logDxnatn(vn)log(DxnbsΠcisiur)Fiberlog(Datnxn(bsΠcisiur))1Fiberabsentnormsubscript𝐷subscript𝑥𝑛superscript𝑎subscript𝑡𝑛subscript𝑣𝑛subscriptnormsubscript𝐷subscript𝑥𝑛superscript𝑏𝑠Πsuperscriptsubscript𝑐𝑖subscript𝑠𝑖superscript𝑢𝑟Fibersubscriptnormsuperscriptsubscript𝐷superscript𝑎subscript𝑡𝑛subscript𝑥𝑛superscript𝑏𝑠Πsuperscriptsubscript𝑐𝑖subscript𝑠𝑖superscript𝑢superscript𝑟1Fiber\displaystyle\quad\geq\log\|D_{x_{n}}a^{t_{n}}\big{(}v_{n}\big{)}\|-\log\|\big{(}D_{x_{n}}b^{s}\Pi c_{i}^{s_{i}}u^{r}\big{)}\|_{\text{Fiber}}-\log\|\big{(}D_{a^{t_{n}}\cdot x_{n}}\big{(}b^{s}\Pi c_{i}^{s_{i}}u^{r^{\prime}}\big{)}\big{)}^{-1}\|_{\text{Fiber}}\phantom{\Big{\|}}
logDxnatn(vn)logDxn(ur)FiberlogDurxn(bsΠcisi)Fiberabsentnormsubscript𝐷subscript𝑥𝑛superscript𝑎subscript𝑡𝑛subscript𝑣𝑛subscriptnormsubscript𝐷subscript𝑥𝑛superscript𝑢𝑟Fibersubscriptnormsubscript𝐷superscript𝑢𝑟subscript𝑥𝑛superscript𝑏𝑠Πsuperscriptsubscript𝑐𝑖subscript𝑠𝑖Fiber\displaystyle\quad\geq\log\|D_{x_{n}}a^{t_{n}}\big{(}v_{n}\big{)}\|-\log\|D_{x_{n}}\big{(}u^{r}\big{)}\|_{\text{Fiber}}-\log\|D_{u^{r}x_{n}}\big{(}b^{s}\Pi c_{i}^{s_{i}}\big{)}\|_{\text{Fiber}}
log(Duratnxn(bsΠcisi))1FiberlogDatnxn(ur)Fiber.subscriptnormsuperscriptsubscript𝐷superscript𝑢superscript𝑟superscript𝑎subscript𝑡𝑛subscript𝑥𝑛superscript𝑏𝑠Πsuperscriptsubscript𝑐𝑖subscript𝑠𝑖1Fibersubscriptnormsubscript𝐷superscript𝑎subscript𝑡𝑛subscript𝑥𝑛superscript𝑢superscript𝑟Fiber\displaystyle\quad\quad\quad-\log\|\big{(}D_{u^{r^{\prime}}a^{t_{n}}\cdot x_{n}}\big{(}b^{s}\Pi c_{i}^{s_{i}}\big{)}\big{)}^{-1}\|_{\text{Fiber}}-\log\|D_{a^{t_{n}}\cdot x_{n}}\big{(}u^{{-r^{\prime}}}\big{)}\|_{\text{Fiber}}.\phantom{\Big{\|}}

Observe that both urxnsuperscript𝑢𝑟subscript𝑥𝑛u^{r}\cdot x_{n} and uratnxnsuperscript𝑢superscript𝑟superscript𝑎subscript𝑡𝑛subscript𝑥𝑛u^{r^{\prime}}a^{t_{n}}\cdot x_{n} are contained in a fixed compact subset of G/Γ𝐺ΓG/\Gamma and hence, by Claim 5.12, having taken δ>0𝛿0\delta>0 sufficiently small in the construction of the Følner sequence, from the constraints on sisubscript𝑠𝑖s_{i} and s𝑠s we have DurxnΠcisibsFibereλtn/100subscriptnormsubscript𝐷superscript𝑢𝑟subscript𝑥𝑛Πsuperscriptsubscript𝑐𝑖subscript𝑠𝑖superscript𝑏𝑠Fibersuperscript𝑒𝜆subscript𝑡𝑛100\|D_{u^{r}x_{n}}\Pi c_{i}^{s_{i}}b^{s}\|_{\text{Fiber}}\leq e^{\lambda t_{n}/100} and (DuratnxnΠcisibs)1Fibereλtn/100subscriptnormsuperscriptsubscript𝐷superscript𝑢superscript𝑟superscript𝑎subscript𝑡𝑛subscript𝑥𝑛Πsuperscriptsubscript𝑐𝑖subscript𝑠𝑖superscript𝑏𝑠1Fibersuperscript𝑒𝜆subscript𝑡𝑛100\|\big{(}D_{u^{r^{\prime}}a^{t_{n}}\cdot x_{n}}\Pi c_{i}^{s_{i}}b^{s}\big{)}^{-1}\|_{\text{Fiber}}\leq e^{\lambda t_{n}/100} for all n𝑛n sufficiently large.

Moreover, from Claim 5.11, we have DxnurFibereλtn/100subscriptnormsubscript𝐷subscript𝑥𝑛superscript𝑢𝑟Fibersuperscript𝑒𝜆subscript𝑡𝑛100\|D_{x_{n}}u^{r}\|_{\text{Fiber}}\leq e^{\lambda t_{n}/100} for all n𝑛n sufficiently large.

Finally, there exists κ>0𝜅0\kappa>0 such that reκtnrnormsuperscript𝑟superscript𝑒𝜅subscript𝑡𝑛norm𝑟\|r^{\prime}\|\leq e^{\kappa t_{n}}\|r\| whence rBm1(e(200+κ)tn)superscript𝑟subscript𝐵superscript𝑚1superscript𝑒200𝜅subscript𝑡𝑛r^{\prime}\in B_{\mathbb{R}^{m-1}}(e^{(200+\kappa)t_{n}}). Again from Claim 5.11, we have DatnxnurFibereλtn/100subscriptnormsubscript𝐷superscript𝑎subscript𝑡𝑛subscript𝑥𝑛superscript𝑢superscript𝑟Fibersuperscript𝑒𝜆subscript𝑡𝑛100\|D_{a^{t_{n}}\cdot x_{n}}u^{-r^{\prime}}\|_{\text{Fiber}}\leq e^{\lambda t_{n}/100} for n𝑛n sufficiently large. Combined with (31) we then have

1tnMαlog𝒜(atn,)dξnλ4100λ.1subscript𝑡𝑛subscriptsuperscript𝑀𝛼norm𝒜superscript𝑎subscript𝑡𝑛𝑑subscript𝜉𝑛𝜆4100𝜆\frac{1}{t_{n}}\int_{M^{\alpha}}{\log\|\mathcal{A}(a^{t_{n}},\cdot)\|\ d\xi_{n}}\geq\lambda-\frac{4}{100}\lambda.

Proposition 5.10 then follows from Lemma 3.10. ∎

5.6. Proof of Proposition 5.1

Having assumed that χmaxsubscript𝜒max\chi_{\mathrm{max}} in (25) is non-zero, we arrive at a contradiction. Take any weak-* subsequential limit μsubscript𝜇\mu_{\infty} of the sequence of measure {μn}subscript𝜇𝑛\{\mu_{n}\} on Mαsuperscript𝑀𝛼M^{\alpha}. We have that μsubscript𝜇\mu_{\infty} is A𝐴A-invariant and has a non-zero fiberwise Lyapunov exponent for the fiberwise derivative over the action of atsuperscript𝑎𝑡a^{t}. Moreover, we have that μsubscript𝜇\mu_{\infty} projects to νsubscript𝜈\nu_{\infty} on G/Γ𝐺ΓG/\Gamma which, as discussed above, is the Haar measure on G/Γ𝐺ΓG/\Gamma. We may replace μsubscript𝜇\mu_{\infty} with an A𝐴A-ergodic component μsuperscript𝜇\mu^{\prime} with the same properties as above. Then μ𝜇\mu is A𝐴A-ergodic, projects to Haar, and the fiberwise derivative cocycle over the A𝐴A-action on (Mα,μ)superscript𝑀𝛼𝜇(M^{\alpha},\mu) has a non-zero Lyapunov exponent functional λi:A.:subscript𝜆𝑖𝐴\lambda_{i}\colon A\to\mathbb{R}.

As in the conclusion of Lemma 4.4, the arguments of [BFH, Section 5.5] using [BRHW, Proposition 5.1] imply that the measure μ𝜇\mu is, in fact, SL(m,)SL𝑚\mathrm{SL}(m,\mathbb{R})-invariant. As before, we note that [BRHW, Proposition 5.1] does not assume ΓΓ\Gamma is cocompact, so the algebraic argument applying that proposition in [BFH, Section 5.5] goes through verbatim. For a more self-contained proof that applies since we only consider the case of SL(m,)SL𝑚\mathrm{SL}(m,\mathbb{R}) see [BDZ, Proposition 4]. We then obtain a contradiction with Zimmer’s cocycle superrigidity by constraints on the dimension of the fibers of Mαsuperscript𝑀𝛼M^{\alpha}. Thus we must have χmax=0subscript𝜒max0\chi_{\mathrm{max}}=0 and Proposition 5.1 follows.

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